Workshop 1 - Linear models
22 novembre, 2019
Links:
Examples of Stan code for regression models
What is in the stan fit object?
1 Data set Howell
Dataset Howell from the rethinking package.
data(Howell1)
howell1 <- Howell1
head(howell1)## height weight age male
## 1 151.765 47.82561 63 1
## 2 139.700 36.48581 63 0
## 3 136.525 31.86484 65 0
## 4 156.845 53.04191 41 1
## 5 145.415 41.27687 51 0
## 6 163.830 62.99259 35 1str(howell1)## 'data.frame': 544 obs. of 4 variables:
## $ height: num 152 140 137 157 145 ...
## $ weight: num 47.8 36.5 31.9 53 41.3 ...
## $ age : num 63 63 65 41 51 35 32 27 19 54 ...
## $ male : int 1 0 0 1 0 1 0 1 0 1 ...ggplot(howell1, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) +
theme(axis.line = element_line(size=1, colour = "black"),
panel.grid.major = element_line(colour = "#d3d3d3"),
panel.grid.minor = element_blank())
Subset with only individuals over 18 years old:
howell2 <- howell1[howell1$age >= 18 , ]plot_grid(ggplot(howell2, aes(height),bins=50) + geom_histogram(aes(y = ..count..), fill="darkgreen",binwidth = 1, alpha=0.4) +
theme(axis.line = element_line(size=1, colour = "black"),
panel.grid.major = element_line(colour = "#d3d3d3"),
panel.grid.minor = element_blank()),
ggplot(howell2, aes(weight),bins=50) + geom_histogram(aes(y = ..count..), fill="blue",binwidth = 1, alpha=0.4) +
theme(axis.line = element_line(size=1, colour = "black"),
panel.grid.major = element_line(colour = "#d3d3d3"),
panel.grid.minor = element_blank()))
ggplot(howell2,aes(y=height,x=weight))+geom_point() + theme(axis.line = element_line(size=1, colour = "black"),
panel.grid.major = element_line(colour = "gray95"),
panel.grid.minor = element_blank())
2 Without predictor
2.1 With large variance
\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,20)\]
Data in a list:
m1.data <- list(N = nrow(howell2), height = howell2$height)2.1.1 Notation 1 (vectorized with target += notation)
explicitly incrementing the log probability function
https://mc-stan.org/docs/2_18/reference-manual/increment-log-prob-section.html
“The basis of Stan’s execution is the evaluation of a log probability function (specifically, a probability density function) for a given set of (real-valued) parameters; this function returns the log density of the posterior up to an additive constant. Data and transformed data are fixed before the log density is evaluated. The total log probability is initialized to zero. Next, any log Jacobian adjustments accrued by the variable constraints are added to the log density (the Jacobian adjustment may be skipped for optimization). Sampling and log probability increment statements may add to the log density in the model block. A log probability increment statement directly increments the log density with the value of an expression as follows.”
https://stackoverflow.com/questions/40289457/stan-using-target-syntax
“The target density is the density from which the sampler samples and it needs to be equal to the joint density of all the parameters given the data up to a constant (which is usually achieved via Bayes’s rule by coding as the joint density of parameters and modeled data up to a constant).”
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
}
parameters { // unobserved variables
real mu;
real<lower=0,upper=50> sigma;
}
model {
target += normal_lpdf(height | mu, sigma); // likelihood
target += normal_lpdf(mu | 178, 20); // prior of the mean
}fit.m1.1 <- sampling(m1.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.1, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: bb69f84792e053a2e0234193f5617c02.
## 2 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=1000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## mu 154.64 0.02 0.41 154.08 154.63 155.18 612 1
## sigma 7.77 0.01 0.29 7.42 7.75 8.14 875 1
## lp__ -1223.11 0.04 0.94 -1224.36 -1222.83 -1222.23 467 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:45:12 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).2.1.2 Notation 2 (vectorized with Jags-like notation)
using the sampling statement
Difference with explicitly incrementing the log probability function:
“The sampling statement drops all the terms in the log probability function that are constant, whereas the explicit call to normal_lpdf adds all of the terms in the definition of the log normal probability function, including all of the constant normalizing terms. Therefore, the explicit increment form can be used to recreate the exact log probability values for the model. Otherwise, the sampling statement form will be faster if any of the input expressions involve only constants, data variables, and transformed data variables.”
https://mc-stan.org/docs/2_18/reference-manual/sampling-statements-section.html
Bounding sigma in the parameter block:
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
}
parameters { // unobserved variables
real mu;
real<lower=0,upper=50> sigma;
}
model {
height ~ normal(mu,sigma); // likelihood
mu ~ normal(178, 20); // prior of the mean
}fit.m1.2.1 <- sampling(m1.2.1, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.1, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 362449507f9771e1464afdb2ddaff73e.
## 2 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=1000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## mu 154.62 0.01 0.41 154.12 154.62 155.17 892 1
## sigma 7.77 0.01 0.29 7.43 7.77 8.17 922 1
## lp__ -895.73 0.04 0.91 -897.01 -895.42 -894.86 542 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:46:01 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).Specifying the prior of sigma in the model block (\(\sigma \sim \mathcal{U}(0,50)\)):
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
}
parameters { // unobserved variables
real mu;
real sigma;
}
model {
height ~ normal(mu,sigma); // likelihood
mu ~ normal(178, 20); // prior of the mean
sigma ~ uniform(0,50); // prior of sigma
}fit.m1.2.2 <- sampling(m1.2.2, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.2.2, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 4ad4544d00fe0bd73bdaf4a55e08b243.
## 2 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=1000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## mu 154.62 0.02 0.41 154.10 154.61 155.15 548 1
## sigma 7.78 0.01 0.30 7.40 7.77 8.17 551 1
## lp__ -897.63 0.05 1.01 -898.94 -897.30 -896.73 422 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:46:47 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).2.1.3 Notation 3 (unvectorized)
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
}
parameters { // unobserved variables
real mu;
real<lower=0,upper=50> sigma;
}
model {
for (n in 1:N)
height[n] ~ normal(mu,sigma); // likelihood
mu ~ normal(178, 20); // prior of the mean
}fit.m1.3 <- sampling(m1.3, data = m1.data, iter = 1000, chains = 2, cores = 2)
print(fit.m1.3, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 2c0d3bdaed726863b2342ed270e93080.
## 2 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=1000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## mu 154.60 0.02 0.42 154.06 154.60 155.14 721 1
## sigma 7.77 0.01 0.30 7.40 7.76 8.18 776 1
## lp__ -895.76 0.04 0.94 -897.04 -895.48 -894.87 456 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:47:37 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).2.1.3.1 Comparing model runtime
Vectorized with target += notation:
get_elapsed_time(fit.m1.1)## warmup sample
## chain:1 0.013483 0.011086
## chain:2 0.018512 0.020148Vectorized with Jags-like notation:
# not specifying the prior of sigma
get_elapsed_time(fit.m1.2.1)## warmup sample
## chain:1 0.020005 0.021851
## chain:2 0.018100 0.012708# Specifying the prior of sigma
get_elapsed_time(fit.m1.2.2)## warmup sample
## chain:1 0.062646 0.015833
## chain:2 0.021128 0.010710Unvectorized:
get_elapsed_time(fit.m1.3)## warmup sample
## chain:1 0.074360 0.090645
## chain:2 0.099856 0.0952812.2 With low variance
\[ h_{i} \sim \mathcal{N}(\mu \, , \, \sigma^2)\] \[\mu \sim \mathcal{N}(178,0.1)\]
Data in a list:
m2.data <- list(N = nrow(howell2), height = howell2$height)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
}
parameters { // unobserved variables
real mu;
real<lower=0,upper=50> sigma;
}
model {
target += normal_lpdf(height | mu, sigma); // likelihood
target += normal_lpdf(mu | 178, 0.1); // prior of the mean
}fit.m2 <- sampling(m2 , data = m2.data, iter = 1000, chains = 2, cores = 2)
print(fit.m2, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 732cdcf02bdc315da3f0b397f4c17596.
## 2 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=1000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## mu 177.86 0.00 0.09 177.74 177.86 177.99 779 1
## sigma 24.56 0.03 0.94 23.37 24.57 25.74 827 1
## lp__ -1623.61 0.05 0.93 -1624.84 -1623.32 -1622.79 365 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:48:28 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).3 With one continious predictor
Model equation:
\[\begin{align*} h_{i} \sim & \, \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} = & \, \beta_{1} + \beta_{2} w_{i}\\[4pt] \beta_{1} \sim & \, \mathcal{N}(178,100)\\[4pt] \beta_{2} \sim & \, \mathcal{N}(0,10)\\[4pt] \sigma \sim & \, \mathcal{U}(0,5)\\[4pt] \end{align*}\]
Centering the predictor:
howell2$weight.c <- howell2$weight - mean(howell2$weight)Data in a list:
m3.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slopes
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight ,sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ normal(0, 10); // prior of the slope
sigma ~ uniform(0,50);
}fit.m3.1 <- sampling(m3.1, data = m3.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: dca6bb84ed30c87a7015873cc74526bf.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 154.60 0.01 0.29 154.23 154.59 154.96 1841 1
## beta[2] 0.90 0.00 0.04 0.85 0.90 0.96 372 1
## sigma 5.11 0.01 0.20 4.87 5.10 5.37 1438 1
## lp__ -749.13 0.05 1.33 -750.77 -748.77 -747.87 600 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:49:21 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).Another possibility for \(\beta_{2}\) prior: \[\beta_{2} \sim \, \text{LogNormal}(0,10)\].
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slopes
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight ,sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ lognormal(0, 1); // prior of the slope
sigma ~ uniform(0,50);
}fit.m3.2 <- sampling(m3.2, data = m3.data, iter = 1000, chains = 4, cores = 4)## Warning: There were 187 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Warning: There were 2 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low## Warning: Examine the pairs() plot to diagnose sampling problemsprint(fit.m3.2, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 76e2c520e9a282812337c63011f1f5e3.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 153.38 1.49 10.27 153.99 154.59 155.09 48 1.06
## beta[2] 0.90 0.00 0.11 0.82 0.90 0.98 978 1.00
## sigma 9.82 3.22 9.68 4.94 5.21 15.10 9 1.36
## lp__ -844.87 66.89 193.61 -975.39 -749.26 -747.81 8 1.36
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:50:06 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).vectorized with target +=
data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slopes
real sigma;
}
model {
target += normal_lpdf(height | beta[1] + beta[2]*weight, sigma); // likelihood
target += normal_lpdf(beta[1] | 178, 20); // prior of the intercept
target += lognormal_lpdf(beta[2] |0, 1); // prior of the slope
target += uniform_lpdf(sigma|0,50);
}fit.m3.3 <- sampling(m3.3, data = m3.data, iter = 1000, chains = 4, cores = 4)## Warning: There were 123 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low## Warning: Examine the pairs() plot to diagnose sampling problemsprint(fit.m3.3, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: c0bd04f9d695ff94776043871bd4d625.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 154.60 0.02 0.67 154.00 154.60 155.17 1102 1.00
## beta[2] 0.89 0.01 0.09 0.79 0.89 0.97 164 1.06
## sigma 9.47 5.34 7.62 4.90 5.19 22.09 2 8.04
## lp__ -1170.98 109.09 154.85 -1433.29 -1082.10 -1080.69 2 13.21
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:50:52 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).3.1 What do our priors imply?
Let’s simulate the prior predictive distribution (simulate heights from the model, using only the priors).
See Chapter 4 of Statistical Rethinking.
set.seed(2971)
N <- 100 # 100 lines
a <- rnorm( N , 178 , 20 )
b1 <- rnorm( N , 0 , 10 )
b2 <- rlnorm( N, 0 , 1 )
dens(rnorm( 1e4, 0 , 10 ),main="b ~ dnorm(0,10)")
dens(rlnorm( 1e4 , 0 , 1 ),main="log(b) ~ dnorm(0,1)")
p <- plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b1[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )
McElreath: “For reference, I’ve added a dashed line at zero—no one is shorter than zero—and the “Wadlow” line at 272cm for the world’s tallest person. The pattern doesn’t look like any human population at all. It essentially says that the relationship between weight and height could be absurdly positive or negative. Before we’ve even seen the data, this is a bad model."
plot( NULL , xlim=range(howell2$weight) , ylim=c(-100,400) ,xlab="weight" , ylab="height")
abline(h=0,lty=2)
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "log(b) ~ dnorm(0,1)" )
xbar <- mean(howell2$weight)
for ( i in 1:N ) curve( a[i] + b2[i]*(x - xbar) ,
from=min(howell2$weight) , to=max(howell2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )
McElreath: “This is much more sensible. There is still a rare impossible relationship. But nearly all lines in the joint prior for \(\beta_{1}\) and \(\beta_{2}\) are now within human reason. We’re fussing about this prior, even though as you’ll see in the next section there is so much data in this example that the priors end up not mattering. We fuss for two reasons. First, there are many analyses in which no amount of data makes the prior irrelevant. In such cases, non-Bayesian procedures are no better off. They also depend upon structural features of the model. Paying careful attention to those features is essential. Second, thinking about the priors helps us develop better models, maybe even eventually going beyond geocentrism.”
Help to choose your prior with Stan: https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations
3.2 Convergence diagnostics
https://github.com/stan-dev/stan/wiki/Stan-Best-Practices
3.2.1 Rhat < 1.1
“The best generic diagnostic for the accuracy of a Markov chain Monte Carlo algorithm is the split R-hat statistic which quantifies the consistency of an ensemble of Markov chains. The idea is to run multiple Markov chains, 4 is an okay default but running more makes the diagnostic more sensitive, and ensure that they’re all exploring the same regions of parameter space. If even one chain is inconsistent with the others then all of the chains are suspect! In practice we have found that requiring Rhat < 1.1 is a good default requirement for each parameter.”
rhats <- rhat(fit.m3.2)
print(rhats)## beta[1] beta[2] sigma lp__
## 1.0578017 0.9999526 1.3590584 1.3611599mcmc_rhat(rhats) + yaxis_text(hjust = 1)
3.2.2 Effective sample size
“Another potential problem to keep in mind is Markov chains that explore very slowly. Slow chains lead to low numbers of effective samples, and if there are too few effective samples then we can’t accurate estimate the number of effective samples at all. A good check for such issues is the number of effective samples per iteration – if N_eff / N < 0.001 then you should be suspect of the effective sample size calculation.”
ratios <- neff_ratio(fit.m3.2)
print(ratios)## beta[1] beta[2] sigma lp__
## 0.023845504 0.488859828 0.004527409 0.004189118mcmc_neff(ratios, size = 2)
3.2.3 MCMC diagnostics using the bayesplot package
We need:
# The posterior draws
posterior <- as.array(fit.m3.2)
# The log of the posterior density for each draw
lp <- log_posterior(fit.m3.2)
head(lp)## Iteration Value Chain
## 1 1 -748.2942 1
## 2 2 -748.4240 1
## 3 3 -747.9331 1
## 4 4 -747.7890 1
## 5 5 -748.0312 1
## 6 6 -751.1590 1# Some NUTS-specific diagnostic values
np <- nuts_params(fit.m3.2)
head(np)## Iteration Parameter Value Chain
## 1 1 accept_stat__ 0.5872795 1
## 2 2 accept_stat__ 0.9966800 1
## 3 3 accept_stat__ 0.9712667 1
## 4 4 accept_stat__ 0.9852274 1
## 5 5 accept_stat__ 0.9135522 1
## 6 6 accept_stat__ 0.7162124 13.2.3.1 Divergent transitions
https://mc-stan.org/bayesplot/articles/visual-mcmc-diagnostics.html
mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma"))
Pairs plots:
mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"), off_diag_args = list(size = 0.75))
Trace plots:
mcmc_trace(posterior, pars = "beta[2]", np = np) + xlab("Post-warmup iteration")
3.3 Posterior predictive distributions
https://mc-stan.org/docs/2_18/stan-users-guide/prediction-forecasting-and-backcasting.html
m4.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.c)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slope
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight ,sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ lognormal(0, 1); // prior of the slope
sigma ~ uniform(0,50);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i], sigma);
}fit.m4 <- sampling(m4, data = m4.data, iter = 1000, chains = 4, cores = 4)
y_rep <- as.matrix(fit.m4, pars = "y_rep")
dim(y_rep)## [1] 2000 352ppc_dens_overlay(howell2$height, y_rep)
plot_grid(ppc_stat(y = howell2$height, yrep = y_rep, stat = "mean"),
ppc_scatter_avg(y = howell2$height, yrep = y_rep))
3.4 Visualizing the variance
See section 4.4 of Statistical rethinking
Model with non-centered weight:
m3.1.bis.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slopes
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight ,sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ normal(0, 10); // prior of the slope
sigma ~ uniform(0,50);
}fit.m3.1.bis <- sampling(m3.1.bis, data = m3.1.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m3.1.bis, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: dca6bb84ed30c87a7015873cc74526bf.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 113.95 0.09 2.08 111.29 113.92 116.54 517 1
## beta[2] 0.90 0.00 0.05 0.85 0.90 0.96 527 1
## sigma 5.10 0.01 0.20 4.86 5.10 5.35 844 1
## lp__ -749.35 0.05 1.31 -751.13 -749.01 -748.05 578 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:52:33 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).3.4.1 Prediction interval for average height
Figure 4.7. of Statistical rethinking
post <- as.data.frame(fit.m3.1.bis)
f_mu <- function(x) post$`beta[1]` + post$`beta[2]` * x
weight_new <- seq(25, 70)
mu <-
sapply(weight_new, f_mu) %>%
tibble::as_tibble() %>%
rename_all(function(x) weight_new) %>%
mutate(Iter = row_number()) %>%
gather(weight, height, -Iter) %>%
group_by(weight) %>%
mutate(hpdi_l = HDInterval::hdi(height, credMass = 0.89)[1],
hpdi_h = HDInterval::hdi(height, credMass = 0.89)[2]) %>%
mutate(mu = mean(height)) %>%
ungroup() %>%
mutate(weight = as.integer(weight))
p <- ggplot()
p1 <- p +
geom_point(data = mu %>% filter(Iter < 101),
aes(weight, height), alpha = .05, color = 'dodgerblue') +
labs(subtitle="Density at each weight")
p2 <- p +
geom_point(data = howell2,
aes(weight, height), shape = 1, color = 'dodgerblue') +
geom_ribbon(data = mu,
aes(x = weight, ymin = hpdi_l, ymax = hpdi_h),
alpha = .2) +
geom_abline(data = post,
aes(intercept = mean(`beta[1]`), slope = mean(`beta[2]`))) +
labs(subtitle="HPDI Interval = 0.89")
grid.arrange(p1, p2, nrow = 1) 
“Left: The first 100 values in the distribution of \(\beta_{1}\) (intercept) at each weight value. Right: The !Kung height data again, now with 89% HPDI of the mean indicated by the shaded region. Compare this region to the distributions of blue points on the left.”
3.4.2 Prediction interval for actual heights
Figure 4.8. of Statistical rethinking
Incorporating the uncertainty related to the standard deviation \(\sigma^{2}\).
sim_ht <-
sapply(weight_new,
function(x)
rnorm(NROW(post),
post$`beta[1]` + post$`beta[2]` * x,
post$sigma)) %>%
as_tibble() %>%
rename_all(function(x) weight_new) %>%
mutate(Iter = row_number()) %>%
gather(weight, height, -Iter) %>%
group_by(weight) %>%
mutate(pi_l = rethinking::PI(height, prob = 0.89)[1],
pi_h = rethinking::PI(height, prob = 0.89)[2]) %>%
ungroup() %>%
mutate(weight = as.integer(weight))
p2 + geom_ribbon(data = sim_ht,
mapping = aes(x=weight, ymin=pi_l, ymax=pi_h), alpha = .1) +
labs(subtitle = 'Prediction Intervals = 0.89')
“Figure 4.8. 89% prediction interval for height, as a function of weight. The solid line is the MAP estimate of the mean height at each weight. The two shaded regions show different 89% plausible regions. The narrow shaded interval around the line is the distribution of \(\beta_{1}\) (intercept) . The wider shaded region represents the region within which the model expects to find 89% of actual heights in the population, at each weight.”"
4 With one categorical predictor
Individual sex is a categorical variable with value 0 whenever the person is a female and a value 1 whenever the person is a male.
ggplot(howell2, aes(x=as.factor(male), y=height, color=as.factor(male))) +
geom_boxplot() + scale_x_discrete(labels=c("0" = "Female", "1" = "Male")) +
theme(axis.title.x=element_blank(),legend.position = "none")
There are two ways to include binary categorical variables, either using indicator variable or index variable. More details in section 5.3.1 of Statistical rethinking (McElreath’s book).
4.1 Indicator variable (or dummy variable)
Using indicator variables is ok with binary variables, but shouldn’t be used for variables with many categories (see 5.3.2 section of Statistical rethiking).
Model equation (like in McElreath’s book):
\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha + \beta_{m} m_{i}\\[4pt] \alpha &\sim \mathcal{N}(178,20)\\[4pt] \beta_{m} &\sim \mathcal{N}(0,10)\\[4pt] \sigma &\sim \mathcal{U}(0,50)\\[4pt] \end{align*}\]
In R, we can convert our categorical variable in a dummy variable like this:
X <- model.matrix(~male, howell2)m5.1.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male)))
str(m5.1.data)## List of 4
## $ N : int 352
## $ height: num [1:352] 152 140 137 157 145 ...
## $ X : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:352] "1" "2" "3" "4" ...
## .. ..$ : chr [1:2] "(Intercept)" "male"
## ..- attr(*, "assign")= int [1:2] 0 1
## $ n : int 2data {
int<lower=0> N; // number of observations
int<lower=0> n; // number of levels of the categorical variable
vector[N] height; // response variables
matrix[N,n] X; // categorical variable
}
parameters {
vector[n] beta; // intercepts of the categorical variable
real sigma;
}
transformed parameters {
vector[N] mu;
mu = X*beta;
}
model {
height ~ normal(mu ,sigma); // likelihood
sigma ~ uniform(0,50);
beta[1] ~ normal(178,20);
beta[2] ~ normal(0,10);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_1 <- sampling(m5.1.1, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_1, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: a89cc41ee4ed1022b476f4484ab5720d.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## beta[2] 10.82 0.03 0.53 10.15 10.82 11.51 392 1.01
## sigma 5.55 0.00 0.21 5.27 5.54 5.82 2099 1.00
## mu[1] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[2] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[3] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[4] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[5] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[6] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[7] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[8] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[9] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[10] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[11] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[12] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[13] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[14] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[15] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[16] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[17] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[18] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[19] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[20] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[21] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[22] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[23] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[24] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[25] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[26] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[27] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[28] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[29] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[30] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[31] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[32] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[33] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[34] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[35] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[36] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[37] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[38] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[39] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[40] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[41] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[42] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[43] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[44] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[45] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[46] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[47] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[48] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[49] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[50] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[51] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[52] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[53] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[54] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[55] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[56] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[57] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[58] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[59] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[60] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[61] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[62] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[63] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[64] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[65] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[66] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[67] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[68] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[69] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[70] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[71] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[72] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[73] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[74] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[75] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[76] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[77] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[78] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[79] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[80] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[81] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[82] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[83] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[84] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[85] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[86] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[87] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[88] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[89] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[90] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[91] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[92] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[93] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[94] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[95] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[96] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[97] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[98] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[99] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[100] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[101] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[102] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[103] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[104] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[105] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[106] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[107] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[108] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[109] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[110] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[111] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[112] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[113] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[114] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[115] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[116] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[117] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[118] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[119] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[120] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[121] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[122] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[123] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[124] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[125] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[126] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[127] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[128] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[129] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[130] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[131] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[132] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[133] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[134] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[135] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[136] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[137] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[138] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[139] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[140] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[141] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[142] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[143] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[144] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[145] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[146] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[147] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[148] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[149] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[150] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[151] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[152] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[153] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[154] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[155] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[156] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[157] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[158] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[159] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[160] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[161] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[162] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[163] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[164] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[165] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[166] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[167] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[168] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[169] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[170] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[171] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[172] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[173] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[174] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[175] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[176] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[177] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[178] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[179] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[180] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[181] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[182] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[183] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[184] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[185] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[186] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[187] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[188] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[189] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[190] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[191] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[192] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[193] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[194] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[195] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[196] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[197] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[198] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[199] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[200] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[201] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[202] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[203] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[204] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[205] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[206] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[207] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[208] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[209] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[210] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[211] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[212] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[213] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[214] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[215] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[216] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[217] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[218] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[219] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[220] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[221] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[222] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[223] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[224] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[225] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[226] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[227] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[228] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[229] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[230] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[231] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[232] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[233] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[234] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[235] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[236] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[237] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[238] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[239] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[240] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[241] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[242] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[243] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[244] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[245] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[246] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[247] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[248] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[249] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[250] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[251] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[252] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[253] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[254] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[255] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[256] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[257] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[258] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[259] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[260] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[261] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[262] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[263] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[264] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[265] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[266] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[267] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[268] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[269] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[270] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[271] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[272] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[273] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[274] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[275] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[276] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[277] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[278] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[279] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[280] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[281] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[282] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[283] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[284] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[285] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[286] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[287] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[288] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[289] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[290] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[291] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[292] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[293] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[294] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[295] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[296] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[297] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[298] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[299] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[300] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[301] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[302] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[303] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[304] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[305] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[306] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[307] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[308] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[309] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[310] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[311] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[312] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[313] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[314] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[315] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[316] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[317] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[318] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[319] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[320] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[321] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[322] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[323] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[324] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[325] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[326] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[327] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[328] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[329] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[330] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[331] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[332] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[333] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[334] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[335] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[336] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[337] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[338] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[339] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[340] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[341] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[342] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[343] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[344] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[345] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[346] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[347] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[348] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[349] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[350] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## mu[351] 149.53 0.02 0.38 149.05 149.53 150.02 594 1.01
## mu[352] 160.35 0.01 0.39 159.86 160.35 160.86 777 1.00
## y_rep[1] 160.43 0.13 5.56 153.29 160.31 167.57 1777 1.00
## y_rep[2] 149.55 0.13 5.57 142.44 149.54 156.74 1712 1.00
## y_rep[3] 149.53 0.12 5.52 142.47 149.52 156.76 2041 1.00
## y_rep[4] 160.19 0.13 5.63 153.02 160.31 167.40 1944 1.00
## y_rep[5] 149.29 0.12 5.48 142.35 149.03 156.36 1932 1.00
## y_rep[6] 160.36 0.13 5.66 153.26 160.19 167.56 1905 1.00
## y_rep[7] 149.50 0.12 5.47 142.61 149.52 156.20 1988 1.00
## y_rep[8] 160.38 0.12 5.50 153.32 160.63 167.19 2064 1.00
## y_rep[9] 149.44 0.13 5.67 142.25 149.29 156.69 1854 1.00
## y_rep[10] 160.43 0.13 5.75 153.26 160.33 168.07 1933 1.00
## y_rep[11] 149.69 0.12 5.55 142.55 149.82 156.65 2163 1.00
## y_rep[12] 160.25 0.12 5.51 153.24 160.19 167.27 2069 1.00
## y_rep[13] 149.45 0.12 5.65 142.31 149.25 156.78 2086 1.00
## y_rep[14] 149.55 0.12 5.49 142.54 149.50 156.52 1989 1.00
## y_rep[15] 149.37 0.13 5.70 142.21 149.36 156.89 2056 1.00
## y_rep[16] 160.36 0.13 5.50 153.15 160.43 167.48 1772 1.00
## y_rep[17] 160.42 0.13 5.52 153.22 160.43 167.63 1690 1.00
## y_rep[18] 149.33 0.13 5.63 142.09 149.47 156.41 1935 1.00
## y_rep[19] 160.13 0.13 5.66 152.96 160.27 167.39 2028 1.00
## y_rep[20] 149.50 0.12 5.49 142.40 149.66 156.47 2013 1.00
## y_rep[21] 160.31 0.13 5.44 153.19 160.44 167.30 1780 1.00
## y_rep[22] 149.26 0.12 5.57 142.06 149.41 156.31 2002 1.00
## y_rep[23] 149.66 0.13 5.61 142.46 149.71 156.92 1962 1.00
## y_rep[24] 160.23 0.12 5.55 153.44 160.16 167.23 2031 1.00
## y_rep[25] 160.47 0.13 5.57 153.24 160.63 167.76 1980 1.00
## y_rep[26] 149.51 0.13 5.62 142.39 149.54 156.64 1899 1.00
## y_rep[27] 149.70 0.14 5.65 142.53 149.71 156.92 1719 1.00
## y_rep[28] 149.41 0.12 5.55 142.34 149.45 156.43 1984 1.00
## y_rep[29] 149.80 0.14 5.58 142.46 149.90 157.04 1633 1.00
## y_rep[30] 149.70 0.13 5.44 142.78 149.73 156.62 1800 1.00
## y_rep[31] 149.54 0.13 5.49 142.53 149.52 156.63 1709 1.00
## y_rep[32] 160.27 0.13 5.64 153.03 160.16 167.65 1758 1.00
## y_rep[33] 160.35 0.13 5.73 153.06 160.19 167.81 1930 1.00
## y_rep[34] 160.02 0.13 5.50 152.80 160.11 166.93 1934 1.00
## y_rep[35] 160.24 0.13 5.42 153.35 160.33 167.01 1849 1.00
## y_rep[36] 149.55 0.12 5.49 142.44 149.56 156.72 2217 1.00
## y_rep[37] 149.57 0.14 5.52 142.49 149.48 156.61 1633 1.00
## y_rep[38] 160.46 0.12 5.47 153.25 160.48 167.28 2041 1.00
## y_rep[39] 149.65 0.12 5.57 142.37 149.66 156.85 2042 1.00
## y_rep[40] 160.11 0.13 5.65 152.87 160.19 167.26 2016 1.00
## y_rep[41] 160.33 0.12 5.55 153.04 160.40 167.53 2044 1.00
## y_rep[42] 160.49 0.12 5.49 153.46 160.56 167.40 2114 1.00
## y_rep[43] 149.61 0.12 5.59 142.67 149.52 156.86 2076 1.00
## y_rep[44] 149.52 0.13 5.64 142.13 149.64 156.56 2012 1.00
## y_rep[45] 149.60 0.13 5.41 142.63 149.54 156.81 1844 1.00
## y_rep[46] 149.54 0.13 5.58 142.49 149.50 156.80 1792 1.00
## y_rep[47] 160.54 0.14 5.54 153.54 160.53 167.81 1625 1.00
## y_rep[48] 149.42 0.12 5.58 142.04 149.60 156.59 2043 1.00
## y_rep[49] 160.32 0.13 5.55 153.51 160.17 167.51 1867 1.00
## y_rep[50] 149.69 0.12 5.53 142.41 149.70 156.79 2010 1.00
## y_rep[51] 149.43 0.13 5.73 141.94 149.44 156.60 2003 1.00
## y_rep[52] 149.29 0.13 5.72 141.63 149.28 156.49 2009 1.00
## y_rep[53] 160.31 0.12 5.45 153.27 160.33 167.33 2007 1.00
## y_rep[54] 160.35 0.13 5.62 153.27 160.38 167.57 1941 1.00
## y_rep[55] 149.60 0.13 5.52 142.65 149.50 157.01 1877 1.00
## y_rep[56] 160.29 0.12 5.45 153.17 160.26 167.04 1903 1.00
## y_rep[57] 149.47 0.13 5.53 142.35 149.46 156.58 1798 1.00
## y_rep[58] 160.25 0.13 5.62 152.84 160.35 167.23 1877 1.00
## y_rep[59] 149.68 0.12 5.64 142.44 149.75 156.78 2171 1.00
## y_rep[60] 160.30 0.12 5.33 153.58 160.26 167.00 2057 1.00
## y_rep[61] 149.49 0.12 5.69 142.19 149.52 156.81 2087 1.00
## y_rep[62] 149.74 0.13 5.65 142.40 149.74 156.82 1814 1.00
## y_rep[63] 160.30 0.14 5.63 153.43 160.19 167.51 1723 1.00
## y_rep[64] 160.42 0.13 5.65 152.99 160.54 167.66 1989 1.00
## y_rep[65] 149.55 0.12 5.50 142.41 149.44 156.81 1951 1.00
## y_rep[66] 160.57 0.12 5.50 153.55 160.68 167.44 1991 1.00
## y_rep[67] 149.66 0.12 5.45 142.81 149.51 156.55 1964 1.00
## y_rep[68] 149.72 0.13 5.74 142.31 149.79 157.17 1888 1.00
## y_rep[69] 160.39 0.13 5.60 153.08 160.50 167.62 1872 1.00
## y_rep[70] 149.59 0.13 5.72 142.09 149.81 156.50 1960 1.00
## y_rep[71] 160.26 0.13 5.52 153.09 160.25 167.36 1944 1.00
## y_rep[72] 160.35 0.13 5.63 153.22 160.29 167.54 1827 1.00
## y_rep[73] 160.49 0.13 5.48 153.47 160.42 167.51 1906 1.00
## y_rep[74] 149.59 0.14 5.77 142.27 149.52 156.97 1669 1.00
## y_rep[75] 149.50 0.13 5.49 142.41 149.42 156.56 1904 1.00
## y_rep[76] 149.45 0.12 5.28 142.72 149.47 156.18 1924 1.00
## y_rep[77] 149.68 0.13 5.58 142.70 149.71 156.88 1811 1.00
## y_rep[78] 160.36 0.13 5.61 153.25 160.41 167.46 1866 1.00
## y_rep[79] 149.86 0.13 5.54 142.55 149.89 156.95 1884 1.00
## y_rep[80] 149.54 0.12 5.49 142.48 149.57 156.65 1981 1.00
## y_rep[81] 160.43 0.12 5.41 153.61 160.52 167.20 2023 1.00
## y_rep[82] 160.23 0.13 5.56 153.04 160.31 167.35 1898 1.00
## y_rep[83] 149.56 0.13 5.59 142.71 149.60 156.51 1976 1.00
## y_rep[84] 160.48 0.13 5.54 153.37 160.52 167.46 1830 1.00
## y_rep[85] 149.45 0.12 5.51 142.35 149.44 156.51 1985 1.00
## y_rep[86] 160.32 0.13 5.45 153.18 160.36 167.18 1843 1.00
## y_rep[87] 149.54 0.13 5.68 142.41 149.54 156.78 2050 1.00
## y_rep[88] 160.42 0.12 5.38 153.51 160.37 167.63 2177 1.00
## y_rep[89] 149.50 0.12 5.39 142.61 149.57 156.49 1911 1.00
## y_rep[90] 160.26 0.13 5.57 153.11 160.19 167.54 1784 1.00
## y_rep[91] 160.10 0.13 5.62 152.88 160.10 167.38 1959 1.00
## y_rep[92] 160.52 0.13 5.56 153.31 160.46 167.78 1972 1.00
## y_rep[93] 149.37 0.12 5.43 142.59 149.35 156.30 2008 1.00
## y_rep[94] 149.83 0.13 5.58 142.88 149.80 157.03 1942 1.00
## y_rep[95] 160.48 0.13 5.52 153.16 160.38 167.63 1936 1.00
## y_rep[96] 160.24 0.14 5.63 152.83 160.29 167.51 1644 1.00
## y_rep[97] 149.39 0.12 5.46 142.46 149.31 156.46 1924 1.00
## y_rep[98] 160.45 0.13 5.45 153.46 160.36 167.25 1882 1.00
## y_rep[99] 160.32 0.13 5.55 153.25 160.28 167.65 1854 1.00
## y_rep[100] 149.62 0.13 5.62 142.55 149.64 156.85 1988 1.00
## y_rep[101] 160.42 0.12 5.55 153.30 160.41 167.40 2017 1.00
## y_rep[102] 149.49 0.13 5.56 142.07 149.58 156.56 1837 1.00
## y_rep[103] 149.43 0.12 5.45 142.38 149.43 156.55 2231 1.00
## y_rep[104] 160.36 0.13 5.46 153.28 160.31 167.39 1643 1.00
## y_rep[105] 160.43 0.12 5.57 153.58 160.29 167.55 1994 1.00
## y_rep[106] 149.29 0.12 5.56 142.19 149.38 156.52 2155 1.00
## y_rep[107] 149.38 0.12 5.60 141.98 149.51 156.26 2062 1.00
## y_rep[108] 149.52 0.12 5.56 142.54 149.50 156.80 2069 1.00
## y_rep[109] 160.26 0.13 5.49 153.56 160.13 167.46 1894 1.00
## y_rep[110] 149.46 0.12 5.56 142.21 149.40 156.54 1988 1.00
## y_rep[111] 160.44 0.13 5.65 152.97 160.50 167.76 2032 1.00
## y_rep[112] 149.75 0.12 5.52 142.73 149.58 156.83 1968 1.00
## y_rep[113] 149.45 0.13 5.63 142.08 149.55 156.56 1880 1.00
## y_rep[114] 160.42 0.13 5.66 153.13 160.20 167.68 1859 1.00
## y_rep[115] 160.14 0.12 5.49 152.98 160.20 167.05 2131 1.00
## y_rep[116] 149.48 0.13 5.75 142.29 149.39 156.86 1956 1.00
## y_rep[117] 160.27 0.13 5.57 153.30 160.31 167.30 1939 1.00
## y_rep[118] 160.30 0.12 5.44 153.55 160.24 167.26 1934 1.00
## y_rep[119] 149.36 0.12 5.48 142.41 149.33 156.25 1958 1.00
## y_rep[120] 149.43 0.13 5.65 142.35 149.34 156.70 1846 1.00
## y_rep[121] 160.26 0.13 5.59 153.00 160.37 167.33 1974 1.00
## y_rep[122] 149.48 0.13 5.71 142.33 149.54 156.63 1978 1.00
## y_rep[123] 160.15 0.13 5.68 152.81 160.23 167.39 1852 1.00
## y_rep[124] 149.51 0.12 5.46 142.51 149.46 156.57 2063 1.00
## y_rep[125] 149.52 0.13 5.44 142.75 149.57 156.34 1760 1.00
## y_rep[126] 149.36 0.13 5.62 142.20 149.27 156.48 2005 1.00
## y_rep[127] 149.53 0.13 5.73 142.30 149.59 156.97 1884 1.00
## y_rep[128] 160.26 0.13 5.58 152.86 160.29 167.49 1978 1.00
## y_rep[129] 149.58 0.13 5.34 142.28 149.65 156.29 1656 1.00
## y_rep[130] 149.57 0.12 5.46 142.57 149.59 156.62 1990 1.00
## y_rep[131] 160.34 0.13 5.65 153.25 160.32 167.62 1883 1.00
## y_rep[132] 160.24 0.12 5.52 153.32 160.05 167.38 2074 1.00
## y_rep[133] 149.64 0.13 5.53 142.54 149.63 156.96 1830 1.00
## y_rep[134] 160.53 0.12 5.60 153.20 160.52 167.48 2068 1.00
## y_rep[135] 149.36 0.12 5.62 141.94 149.43 156.45 2088 1.00
## y_rep[136] 160.41 0.13 5.76 153.26 160.42 167.77 1930 1.00
## y_rep[137] 160.28 0.13 5.71 153.15 160.10 167.60 1987 1.00
## y_rep[138] 160.32 0.13 5.38 153.38 160.27 167.07 1672 1.00
## y_rep[139] 149.52 0.13 5.67 142.16 149.48 156.88 1989 1.00
## y_rep[140] 160.43 0.13 5.59 153.33 160.35 167.56 1718 1.00
## y_rep[141] 149.44 0.12 5.60 142.03 149.40 156.56 2088 1.00
## y_rep[142] 149.74 0.12 5.52 142.49 149.82 156.76 1983 1.00
## y_rep[143] 149.39 0.12 5.52 142.23 149.30 156.42 2033 1.00
## y_rep[144] 149.55 0.12 5.70 142.63 149.45 156.60 2130 1.00
## y_rep[145] 149.52 0.12 5.59 142.20 149.59 156.60 2037 1.00
## y_rep[146] 160.56 0.12 5.59 153.36 160.52 167.48 2056 1.00
## y_rep[147] 149.63 0.13 5.67 142.24 149.64 156.88 1912 1.00
## y_rep[148] 149.38 0.13 5.54 142.20 149.30 156.44 1893 1.00
## y_rep[149] 160.49 0.13 5.64 153.39 160.38 167.54 1984 1.00
## y_rep[150] 149.43 0.12 5.68 142.22 149.20 156.86 2164 1.00
## y_rep[151] 160.69 0.13 5.64 153.47 160.67 168.08 1931 1.00
## y_rep[152] 149.63 0.12 5.54 142.46 149.63 156.92 2036 1.00
## y_rep[153] 160.32 0.13 5.62 153.19 160.51 167.54 1811 1.00
## y_rep[154] 149.43 0.12 5.41 142.71 149.33 156.40 2053 1.00
## y_rep[155] 160.40 0.13 5.71 152.87 160.41 167.52 2017 1.00
## y_rep[156] 149.52 0.12 5.56 142.50 149.63 156.79 2159 1.00
## y_rep[157] 149.33 0.12 5.56 142.42 149.41 156.47 2090 1.00
## y_rep[158] 160.40 0.13 5.53 153.41 160.41 167.48 1904 1.00
## y_rep[159] 160.49 0.12 5.58 153.28 160.62 167.63 2096 1.00
## y_rep[160] 149.37 0.13 5.52 142.35 149.24 156.49 1902 1.00
## y_rep[161] 160.12 0.12 5.54 152.81 160.10 167.29 2019 1.00
## y_rep[162] 149.40 0.13 5.62 142.41 149.35 156.57 1946 1.00
## y_rep[163] 160.43 0.12 5.50 153.57 160.36 167.36 2110 1.00
## y_rep[164] 160.36 0.13 5.65 153.04 160.51 167.60 1850 1.00
## y_rep[165] 160.43 0.12 5.50 153.45 160.39 167.39 1968 1.00
## y_rep[166] 149.38 0.13 5.67 142.21 149.35 156.63 1853 1.00
## y_rep[167] 160.40 0.13 5.79 152.83 160.52 167.76 2036 1.00
## y_rep[168] 149.64 0.13 5.56 142.65 149.67 156.72 1929 1.00
## y_rep[169] 149.49 0.13 5.62 142.25 149.33 156.79 1980 1.00
## y_rep[170] 149.60 0.13 5.52 142.48 149.73 156.72 1943 1.00
## y_rep[171] 149.66 0.12 5.49 142.59 149.62 156.67 1985 1.00
## y_rep[172] 160.34 0.14 5.60 153.09 160.52 167.18 1625 1.00
## y_rep[173] 149.20 0.12 5.53 141.92 149.16 156.22 1959 1.00
## y_rep[174] 160.41 0.13 5.63 153.28 160.42 167.68 1803 1.00
## y_rep[175] 149.43 0.12 5.56 142.25 149.33 156.70 2086 1.00
## y_rep[176] 160.32 0.12 5.67 152.95 160.32 167.54 2097 1.00
## y_rep[177] 149.44 0.12 5.62 142.26 149.46 156.63 2077 1.00
## y_rep[178] 149.49 0.12 5.57 142.75 149.57 156.45 2023 1.00
## y_rep[179] 160.44 0.13 5.69 153.16 160.47 167.78 2070 1.00
## y_rep[180] 149.38 0.13 5.68 142.01 149.29 156.73 1994 1.00
## y_rep[181] 160.47 0.13 5.66 153.18 160.49 167.61 2015 1.00
## y_rep[182] 149.53 0.13 5.70 142.29 149.30 157.14 1844 1.00
## y_rep[183] 149.72 0.13 5.56 142.75 149.77 156.83 1876 1.00
## y_rep[184] 149.66 0.12 5.48 142.73 149.50 156.68 1973 1.00
## y_rep[185] 160.28 0.13 5.54 153.14 160.37 167.24 1901 1.00
## y_rep[186] 160.38 0.13 5.48 153.27 160.38 167.51 1914 1.00
## y_rep[187] 149.58 0.13 5.70 142.19 149.58 156.99 1992 1.00
## y_rep[188] 160.32 0.12 5.52 153.36 160.20 167.67 1983 1.00
## y_rep[189] 149.74 0.12 5.54 142.59 149.79 156.83 2031 1.00
## y_rep[190] 160.42 0.13 5.71 153.03 160.54 167.70 1826 1.00
## y_rep[191] 149.57 0.13 5.62 142.39 149.53 156.64 1937 1.00
## y_rep[192] 160.53 0.13 5.47 153.72 160.36 167.49 1791 1.00
## y_rep[193] 149.44 0.13 5.53 142.37 149.45 156.39 1878 1.00
## y_rep[194] 160.35 0.13 5.51 153.44 160.36 167.26 1815 1.00
## y_rep[195] 149.33 0.12 5.47 142.62 149.21 156.58 2149 1.00
## y_rep[196] 149.38 0.12 5.61 142.13 149.43 156.65 2108 1.00
## y_rep[197] 149.52 0.13 5.64 142.15 149.49 156.97 1923 1.00
## y_rep[198] 160.23 0.13 5.70 153.00 160.27 167.57 1936 1.00
## y_rep[199] 160.26 0.12 5.39 153.32 160.22 166.95 2031 1.00
## y_rep[200] 149.59 0.12 5.44 142.27 149.80 156.48 1897 1.00
## y_rep[201] 160.49 0.13 5.54 153.40 160.34 167.49 1896 1.00
## y_rep[202] 149.37 0.12 5.51 142.30 149.34 156.59 2003 1.00
## y_rep[203] 160.36 0.12 5.61 152.97 160.51 167.24 2086 1.00
## y_rep[204] 149.46 0.13 5.47 142.44 149.53 156.69 1758 1.00
## y_rep[205] 160.41 0.13 5.49 153.24 160.47 167.32 1885 1.00
## y_rep[206] 149.42 0.13 5.62 142.34 149.43 156.59 1974 1.00
## y_rep[207] 149.78 0.13 5.60 142.60 149.82 156.95 1906 1.00
## y_rep[208] 149.48 0.13 5.49 142.60 149.38 156.62 1823 1.00
## y_rep[209] 149.59 0.13 5.52 142.60 149.86 156.31 1813 1.00
## y_rep[210] 149.73 0.14 5.61 142.42 149.81 156.89 1706 1.00
## y_rep[211] 160.28 0.13 5.60 153.18 160.18 167.73 1825 1.00
## y_rep[212] 160.23 0.13 5.63 153.13 160.21 167.37 1903 1.00
## y_rep[213] 160.60 0.13 5.74 153.36 160.58 168.05 1983 1.00
## y_rep[214] 149.53 0.14 5.62 142.23 149.44 156.97 1722 1.00
## y_rep[215] 160.28 0.13 5.57 152.99 160.37 167.24 1920 1.00
## y_rep[216] 160.41 0.13 5.76 153.13 160.35 167.70 1958 1.00
## y_rep[217] 160.35 0.13 5.59 153.38 160.31 167.49 1918 1.00
## y_rep[218] 149.75 0.12 5.57 142.65 149.80 156.81 2018 1.00
## y_rep[219] 160.39 0.13 5.41 153.26 160.52 167.13 1682 1.00
## y_rep[220] 149.62 0.13 5.70 142.34 149.60 157.02 1911 1.00
## y_rep[221] 149.44 0.12 5.78 142.10 149.50 156.78 2181 1.00
## y_rep[222] 149.76 0.12 5.44 143.01 149.70 156.83 2000 1.00
## y_rep[223] 160.31 0.12 5.60 152.82 160.43 167.36 2085 1.00
## y_rep[224] 149.44 0.12 5.66 142.49 149.44 156.72 2094 1.00
## y_rep[225] 160.28 0.13 5.60 152.96 160.32 167.42 1964 1.00
## y_rep[226] 149.50 0.12 5.61 142.59 149.41 156.83 2122 1.00
## y_rep[227] 149.27 0.13 5.60 142.17 149.31 156.41 1899 1.00
## y_rep[228] 160.39 0.13 5.78 152.83 160.39 167.93 1953 1.00
## y_rep[229] 149.38 0.13 5.37 142.44 149.38 156.36 1821 1.00
## y_rep[230] 160.43 0.12 5.53 153.31 160.24 167.55 2047 1.00
## y_rep[231] 160.45 0.12 5.61 153.49 160.38 167.67 2036 1.00
## y_rep[232] 149.43 0.12 5.45 142.46 149.39 156.47 1945 1.00
## y_rep[233] 149.63 0.13 5.62 142.36 149.66 156.87 2019 1.00
## y_rep[234] 160.32 0.12 5.56 153.06 160.48 167.31 1995 1.00
## y_rep[235] 160.38 0.13 5.66 153.09 160.40 167.55 1890 1.00
## y_rep[236] 149.62 0.13 5.50 142.68 149.56 156.74 1927 1.00
## y_rep[237] 160.30 0.12 5.55 153.06 160.39 167.34 2107 1.00
## y_rep[238] 149.60 0.13 5.45 142.56 149.63 156.61 1898 1.00
## y_rep[239] 149.43 0.12 5.46 142.22 149.51 156.35 1950 1.00
## y_rep[240] 149.68 0.13 5.57 142.57 149.61 156.84 1957 1.00
## y_rep[241] 149.65 0.13 5.55 142.41 149.65 156.84 1872 1.00
## y_rep[242] 160.25 0.13 5.60 153.08 160.21 167.45 1887 1.00
## y_rep[243] 160.10 0.12 5.48 153.17 160.11 167.16 2222 1.00
## y_rep[244] 149.39 0.13 5.55 142.32 149.28 156.51 1950 1.00
## y_rep[245] 160.33 0.13 5.62 153.37 160.21 167.48 1952 1.00
## y_rep[246] 149.50 0.13 5.55 142.13 149.54 156.55 1946 1.00
## y_rep[247] 160.36 0.13 5.63 153.12 160.19 167.72 2029 1.00
## y_rep[248] 149.51 0.13 5.67 142.44 149.51 156.87 1967 1.00
## y_rep[249] 160.46 0.13 5.67 153.11 160.30 167.91 2030 1.00
## y_rep[250] 160.36 0.13 5.76 152.99 160.41 167.49 1961 1.00
## y_rep[251] 160.48 0.12 5.52 153.48 160.42 167.55 2096 1.00
## y_rep[252] 160.39 0.12 5.61 153.24 160.44 167.60 2136 1.00
## y_rep[253] 149.62 0.13 5.54 142.76 149.52 156.56 1864 1.00
## y_rep[254] 149.49 0.13 5.65 142.28 149.47 156.62 2039 1.00
## y_rep[255] 149.55 0.12 5.41 142.55 149.50 156.51 2104 1.00
## y_rep[256] 149.55 0.13 5.46 142.63 149.52 156.49 1647 1.00
## y_rep[257] 160.19 0.13 5.57 153.09 160.13 167.45 1943 1.00
## y_rep[258] 149.44 0.12 5.57 142.40 149.49 156.43 1997 1.00
## y_rep[259] 160.42 0.12 5.48 153.36 160.40 167.44 2104 1.00
## y_rep[260] 160.40 0.12 5.52 153.49 160.33 167.20 1949 1.00
## y_rep[261] 160.40 0.12 5.60 153.18 160.32 167.62 2064 1.00
## y_rep[262] 149.58 0.12 5.31 142.48 149.51 156.39 2002 1.00
## y_rep[263] 149.47 0.13 5.51 142.41 149.42 156.53 1886 1.00
## y_rep[264] 149.48 0.12 5.61 142.26 149.60 156.47 2131 1.00
## y_rep[265] 160.49 0.12 5.68 153.18 160.53 167.72 2191 1.00
## y_rep[266] 149.58 0.13 5.63 142.49 149.55 156.78 1967 1.00
## y_rep[267] 160.22 0.13 5.71 153.06 160.16 167.38 1888 1.00
## y_rep[268] 149.41 0.13 5.58 142.24 149.38 156.64 1860 1.00
## y_rep[269] 149.57 0.12 5.66 142.21 149.75 156.52 2075 1.00
## y_rep[270] 160.26 0.12 5.46 153.48 160.19 167.34 2000 1.00
## y_rep[271] 160.67 0.12 5.42 153.90 160.71 167.64 2016 1.00
## y_rep[272] 149.57 0.12 5.68 142.43 149.47 156.89 2119 1.00
## y_rep[273] 160.02 0.13 5.58 152.57 160.14 166.98 1906 1.00
## y_rep[274] 160.32 0.13 5.56 153.16 160.26 167.56 1929 1.00
## y_rep[275] 149.54 0.12 5.61 142.49 149.43 156.69 2062 1.00
## y_rep[276] 160.36 0.13 5.62 152.85 160.46 167.57 1923 1.00
## y_rep[277] 149.40 0.13 5.45 142.27 149.51 156.33 1775 1.00
## y_rep[278] 149.53 0.13 5.50 142.49 149.51 156.59 1677 1.00
## y_rep[279] 160.41 0.13 5.63 153.16 160.38 167.59 1951 1.00
## y_rep[280] 149.45 0.12 5.50 142.40 149.54 156.46 1972 1.00
## y_rep[281] 149.39 0.13 5.64 142.32 149.43 156.64 1954 1.00
## y_rep[282] 160.43 0.12 5.46 153.14 160.53 167.40 2203 1.00
## y_rep[283] 149.69 0.12 5.50 142.74 149.65 156.59 2158 1.00
## y_rep[284] 160.38 0.13 5.44 153.46 160.37 167.30 1887 1.00
## y_rep[285] 149.50 0.13 5.50 142.55 149.58 156.48 1876 1.00
## y_rep[286] 160.39 0.13 5.66 153.05 160.51 167.61 1986 1.00
## y_rep[287] 160.46 0.13 5.54 153.35 160.46 167.45 1931 1.00
## y_rep[288] 160.47 0.13 5.57 153.41 160.73 167.28 1898 1.00
## y_rep[289] 149.60 0.12 5.43 142.60 149.62 156.44 1981 1.00
## y_rep[290] 160.19 0.13 5.50 153.14 160.32 166.97 1925 1.00
## y_rep[291] 160.37 0.12 5.51 153.47 160.18 167.44 1989 1.00
## y_rep[292] 160.41 0.12 5.63 153.21 160.47 167.81 2210 1.00
## y_rep[293] 160.59 0.13 5.69 153.08 160.47 167.96 2011 1.00
## y_rep[294] 149.71 0.12 5.60 142.72 149.71 156.96 2016 1.00
## y_rep[295] 160.32 0.12 5.44 153.07 160.53 167.14 2000 1.00
## y_rep[296] 149.40 0.12 5.50 142.47 149.23 156.62 1977 1.00
## y_rep[297] 149.47 0.14 5.69 142.15 149.43 156.77 1610 1.00
## y_rep[298] 149.32 0.13 5.54 142.13 149.31 156.40 1940 1.00
## y_rep[299] 160.51 0.13 5.55 153.43 160.38 167.76 1947 1.00
## y_rep[300] 160.55 0.14 5.60 153.52 160.48 167.86 1668 1.00
## y_rep[301] 149.61 0.12 5.56 142.26 149.67 156.70 2029 1.00
## y_rep[302] 160.17 0.13 5.47 153.12 160.24 166.92 1840 1.00
## y_rep[303] 149.41 0.13 5.58 142.03 149.36 156.43 1843 1.00
## y_rep[304] 160.36 0.14 5.71 153.03 160.38 167.56 1740 1.00
## y_rep[305] 160.38 0.12 5.54 153.28 160.24 167.62 2050 1.00
## y_rep[306] 160.34 0.13 5.64 153.21 160.32 167.48 1756 1.00
## y_rep[307] 160.45 0.12 5.56 153.53 160.39 167.53 2170 1.00
## y_rep[308] 149.52 0.14 5.74 142.28 149.56 156.84 1749 1.00
## y_rep[309] 149.50 0.12 5.55 142.54 149.53 156.76 2010 1.00
## y_rep[310] 149.49 0.13 5.46 142.69 149.34 156.68 1829 1.00
## y_rep[311] 149.34 0.14 5.68 142.31 149.42 156.69 1605 1.00
## y_rep[312] 149.56 0.12 5.55 142.23 149.51 156.67 1973 1.00
## y_rep[313] 160.48 0.12 5.51 153.61 160.38 167.53 2170 1.00
## y_rep[314] 149.48 0.13 5.60 142.41 149.49 156.60 1998 1.00
## y_rep[315] 160.33 0.13 5.60 153.17 160.37 167.50 1894 1.00
## y_rep[316] 149.41 0.12 5.70 141.95 149.32 156.75 2094 1.00
## y_rep[317] 160.37 0.13 5.59 153.06 160.38 167.56 1912 1.00
## y_rep[318] 149.39 0.13 5.63 142.22 149.33 156.74 1825 1.00
## y_rep[319] 160.20 0.13 5.62 152.97 160.12 167.49 1804 1.00
## y_rep[320] 149.46 0.13 5.52 142.31 149.49 156.43 1903 1.00
## y_rep[321] 160.49 0.13 5.54 153.53 160.55 167.43 1864 1.00
## y_rep[322] 149.57 0.12 5.36 142.64 149.67 156.22 1924 1.00
## y_rep[323] 160.36 0.14 5.49 153.29 160.31 167.21 1588 1.00
## y_rep[324] 160.47 0.13 5.50 153.55 160.42 167.56 1913 1.00
## y_rep[325] 149.64 0.13 5.70 142.34 149.67 156.87 1791 1.00
## y_rep[326] 149.77 0.13 5.66 142.21 149.92 156.84 1962 1.00
## y_rep[327] 149.62 0.12 5.58 142.37 149.59 156.87 2166 1.00
## y_rep[328] 160.12 0.13 5.51 152.99 160.14 167.20 1726 1.00
## y_rep[329] 160.36 0.13 5.64 152.95 160.33 167.58 1900 1.00
## y_rep[330] 160.36 0.13 5.75 153.05 160.40 167.93 1990 1.00
## y_rep[331] 160.25 0.13 5.61 153.09 160.29 167.46 2002 1.00
## y_rep[332] 160.25 0.13 5.60 153.09 160.12 167.59 1866 1.00
## y_rep[333] 149.43 0.13 5.70 142.21 149.44 156.79 1879 1.00
## y_rep[334] 160.49 0.12 5.40 153.50 160.42 167.53 1961 1.00
## y_rep[335] 149.63 0.12 5.55 142.45 149.63 156.66 1986 1.00
## y_rep[336] 160.41 0.13 5.52 153.38 160.38 167.52 1864 1.00
## y_rep[337] 149.65 0.13 5.68 142.41 149.60 156.95 2048 1.00
## y_rep[338] 149.20 0.13 5.59 141.71 149.24 156.50 1924 1.00
## y_rep[339] 149.55 0.13 5.65 142.58 149.53 156.68 1801 1.00
## y_rep[340] 149.57 0.12 5.48 142.57 149.53 156.55 2013 1.00
## y_rep[341] 149.49 0.12 5.52 142.34 149.55 156.62 2019 1.00
## y_rep[342] 160.27 0.13 5.52 153.30 160.25 167.38 1797 1.00
## y_rep[343] 149.34 0.13 5.63 142.14 149.36 156.30 1954 1.00
## y_rep[344] 160.33 0.13 5.66 153.21 160.32 167.54 1867 1.00
## y_rep[345] 149.74 0.12 5.42 142.98 149.62 156.83 1979 1.00
## y_rep[346] 160.32 0.12 5.45 153.19 160.32 167.21 1908 1.00
## y_rep[347] 149.46 0.14 5.59 142.19 149.45 156.86 1649 1.00
## y_rep[348] 149.65 0.13 5.57 142.42 149.61 156.93 1861 1.00
## y_rep[349] 149.47 0.13 5.56 142.43 149.46 156.60 1978 1.00
## y_rep[350] 160.22 0.14 5.45 153.38 160.09 167.26 1589 1.00
## y_rep[351] 149.60 0.12 5.45 142.44 149.57 156.65 1914 1.00
## y_rep[352] 160.49 0.13 5.56 153.09 160.68 167.30 1937 1.00
## lp__ -780.37 0.04 1.11 -781.84 -780.08 -779.27 673 1.00
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:53:34 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).summary(lm(height~as.factor(male),data=howell2))##
## Call:
## lm(formula = height ~ as.factor(male), data = howell2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20.6585 -3.4635 0.2965 3.4715 18.7115
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 149.5135 0.4049 369.25 <2e-16 ***
## as.factor(male)1 10.8450 0.5914 18.34 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared: 0.49, Adjusted R-squared: 0.4885
## F-statistic: 336.3 on 1 and 350 DF, p-value: < 2.2e-16Convergence diagnostics
posterior <- as.array(fit.m5_1_1) # posterior draws
lp <- log_posterior(fit.m5_1_1) # log of the posterior density for each draw
np <- nuts_params(fit.m5_1_1) # Some NUTS-specific diagnostic values
mcmc_parcoord(posterior, np = np,pars=c("beta[1]","beta[2]","sigma")) 
mcmc_pairs(posterior, np = np, pars = c("beta[1]","beta[2]","sigma"), off_diag_args = list(size = 0.75)) # Pairs plots
mcmc_trace(posterior, pars = c("beta[1]","beta[2]","sigma"), np = np) + xlab("Post-warmup iteration") # Trace plots## No divergences to plot.
Same outputs with another prior for sigma: \(\sigma \sim \, {\rm Cauchy} (0,5)\)
data {
int<lower=0> N; // number of observations
int<lower=0> n; // number of levels of the categorical variable
vector[N] height; // response variables
matrix[N,n] X; // categorical variable
}
parameters {
vector[n] beta; // intercepts of the categorical variable
real sigma;
}
transformed parameters {
vector[N] mu;
mu = X*beta;
}
model {
height ~ normal(mu ,sigma); // likelihood
sigma ~ cauchy(0,5);
beta[1] ~ normal(178,20);
beta[2] ~ normal(0,10);
// beta ~ normal(0,1000)
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_1_2 <- sampling(m5.1.2, data = m5.1.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_1_2, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 4c192013905ef0bcb3999ba4bc93cd6e.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## beta[2] 10.78 0.02 0.60 10.03 10.77 11.54 1292 1
## sigma 5.54 0.01 0.21 5.26 5.53 5.81 1498 1
## mu[1] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[2] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[3] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[4] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[5] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[6] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[7] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[8] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[9] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[10] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[11] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[12] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[13] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[14] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[15] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[16] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[17] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[18] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[19] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[20] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[21] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[22] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[23] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[24] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[25] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[26] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[27] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[28] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[29] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[30] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[31] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[32] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[33] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[34] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[35] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[36] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[37] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[38] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[39] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[40] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[41] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[42] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[43] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[44] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[45] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[46] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[47] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[48] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[49] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[50] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[51] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[52] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[53] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[54] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[55] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[56] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[57] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[58] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[59] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[60] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[61] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[62] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[63] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[64] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[65] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[66] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[67] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[68] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[69] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[70] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[71] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[72] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[73] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[74] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[75] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[76] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[77] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[78] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[79] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[80] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[81] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[82] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[83] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[84] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[85] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[86] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[87] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[88] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[89] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[90] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[91] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[92] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[93] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[94] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[95] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[96] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[97] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[98] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[99] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[100] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[101] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[102] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[103] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[104] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[105] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[106] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[107] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[108] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[109] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[110] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[111] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[112] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[113] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[114] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[115] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[116] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[117] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[118] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[119] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[120] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[121] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[122] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[123] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[124] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[125] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[126] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[127] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[128] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[129] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[130] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[131] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[132] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[133] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[134] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[135] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[136] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[137] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[138] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[139] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[140] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[141] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[142] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[143] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[144] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[145] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[146] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[147] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[148] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[149] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[150] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[151] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[152] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[153] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[154] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[155] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[156] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[157] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[158] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[159] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[160] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[161] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[162] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[163] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[164] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[165] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[166] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[167] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[168] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[169] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[170] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[171] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[172] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[173] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[174] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[175] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[176] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[177] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[178] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[179] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[180] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[181] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[182] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[183] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[184] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[185] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[186] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[187] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[188] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[189] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[190] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[191] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[192] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[193] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[194] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[195] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[196] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[197] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[198] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[199] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[200] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[201] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[202] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[203] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[204] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[205] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[206] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[207] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[208] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[209] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[210] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[211] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[212] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[213] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[214] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[215] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[216] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[217] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[218] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[219] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[220] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[221] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[222] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[223] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[224] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[225] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[226] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[227] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[228] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[229] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[230] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[231] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[232] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[233] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[234] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[235] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[236] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[237] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[238] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[239] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[240] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[241] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[242] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[243] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[244] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[245] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[246] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[247] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[248] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[249] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[250] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[251] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[252] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[253] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[254] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[255] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[256] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[257] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[258] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[259] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[260] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[261] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[262] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[263] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[264] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[265] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[266] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[267] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[268] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[269] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[270] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[271] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[272] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[273] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[274] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[275] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[276] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[277] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[278] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[279] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[280] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[281] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[282] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[283] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[284] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[285] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[286] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[287] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[288] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[289] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[290] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[291] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[292] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[293] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[294] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[295] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[296] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[297] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[298] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[299] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[300] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[301] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[302] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[303] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[304] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[305] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[306] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[307] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[308] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[309] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[310] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[311] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[312] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[313] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[314] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[315] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[316] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[317] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[318] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[319] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[320] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[321] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[322] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[323] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[324] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[325] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[326] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[327] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[328] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[329] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[330] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[331] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[332] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[333] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[334] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[335] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[336] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[337] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[338] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[339] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[340] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[341] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[342] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[343] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[344] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[345] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[346] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[347] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[348] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[349] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[350] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## mu[351] 149.55 0.01 0.41 149.03 149.55 150.05 1204 1
## mu[352] 160.33 0.01 0.40 159.83 160.33 160.84 2237 1
## y_rep[1] 160.13 0.12 5.59 153.00 160.17 167.24 2073 1
## y_rep[2] 149.77 0.12 5.47 142.76 149.78 156.81 1958 1
## y_rep[3] 149.47 0.12 5.44 142.22 149.53 156.48 1976 1
## y_rep[4] 160.44 0.13 5.70 153.07 160.37 167.93 2051 1
## y_rep[5] 149.75 0.13 5.47 142.70 149.79 156.65 1917 1
## y_rep[6] 160.41 0.12 5.67 153.22 160.39 167.56 2062 1
## y_rep[7] 149.55 0.13 5.40 142.60 149.54 156.47 1800 1
## y_rep[8] 160.45 0.13 5.49 153.62 160.31 167.53 1828 1
## y_rep[9] 149.71 0.14 5.61 142.38 149.72 156.88 1664 1
## y_rep[10] 160.60 0.13 5.71 153.47 160.49 167.83 2067 1
## y_rep[11] 149.55 0.13 5.46 142.45 149.51 156.66 1788 1
## y_rep[12] 160.24 0.12 5.63 152.97 160.31 167.36 2045 1
## y_rep[13] 149.80 0.12 5.35 143.11 150.04 156.64 2064 1
## y_rep[14] 149.32 0.13 5.47 142.29 149.32 156.23 1792 1
## y_rep[15] 149.57 0.12 5.45 142.37 149.59 156.61 1907 1
## y_rep[16] 160.21 0.12 5.57 152.97 160.23 167.31 2010 1
## y_rep[17] 160.16 0.12 5.40 153.21 160.34 166.83 2071 1
## y_rep[18] 149.48 0.13 5.47 142.68 149.34 156.74 1834 1
## y_rep[19] 160.29 0.12 5.59 153.06 160.28 167.45 2043 1
## y_rep[20] 149.44 0.13 5.49 142.48 149.36 156.68 1677 1
## y_rep[21] 160.53 0.13 5.70 153.20 160.61 167.74 2000 1
## y_rep[22] 149.76 0.12 5.39 142.96 149.74 156.67 2057 1
## y_rep[23] 149.69 0.12 5.56 142.42 149.77 156.89 2020 1
## y_rep[24] 160.57 0.12 5.51 153.44 160.57 167.46 1965 1
## y_rep[25] 160.25 0.12 5.41 153.34 160.32 167.28 1982 1
## y_rep[26] 149.69 0.13 5.60 142.48 149.67 156.66 1866 1
## y_rep[27] 149.47 0.13 5.48 142.36 149.58 156.38 1740 1
## y_rep[28] 149.68 0.13 5.59 142.53 149.63 156.77 1733 1
## y_rep[29] 149.78 0.13 5.59 142.46 149.67 156.97 1896 1
## y_rep[30] 149.55 0.12 5.37 142.66 149.64 156.36 2113 1
## y_rep[31] 149.38 0.13 5.52 142.33 149.33 156.43 1807 1
## y_rep[32] 160.32 0.13 5.55 153.33 160.13 167.45 1801 1
## y_rep[33] 160.37 0.13 5.57 153.09 160.50 167.06 1750 1
## y_rep[34] 160.41 0.13 5.78 153.05 160.43 167.79 2103 1
## y_rep[35] 160.12 0.13 5.51 153.10 159.98 167.12 1888 1
## y_rep[36] 149.40 0.13 5.62 142.26 149.31 156.66 1833 1
## y_rep[37] 149.62 0.12 5.49 142.77 149.59 156.62 2163 1
## y_rep[38] 160.48 0.13 5.52 153.27 160.52 167.57 1927 1
## y_rep[39] 149.72 0.12 5.61 142.44 149.61 156.96 2058 1
## y_rep[40] 160.40 0.13 5.66 152.96 160.21 167.70 1952 1
## y_rep[41] 160.21 0.12 5.70 153.16 160.19 167.15 2132 1
## y_rep[42] 160.38 0.13 5.56 153.13 160.34 167.44 1886 1
## y_rep[43] 149.38 0.13 5.72 142.08 149.36 156.43 2017 1
## y_rep[44] 149.39 0.13 5.43 142.24 149.55 156.25 1854 1
## y_rep[45] 149.70 0.12 5.57 142.74 149.62 156.82 1996 1
## y_rep[46] 149.58 0.13 5.68 142.32 149.60 156.83 1865 1
## y_rep[47] 160.25 0.12 5.47 153.23 160.27 167.40 2066 1
## y_rep[48] 149.57 0.12 5.55 142.65 149.53 156.83 1982 1
## y_rep[49] 160.15 0.12 5.61 153.15 160.21 167.38 2031 1
## y_rep[50] 149.57 0.13 5.65 142.12 149.55 156.73 1889 1
## y_rep[51] 149.72 0.13 5.67 142.39 149.62 157.03 1768 1
## y_rep[52] 149.28 0.13 5.66 142.06 149.22 156.52 1903 1
## y_rep[53] 160.22 0.12 5.49 153.23 160.32 167.14 2028 1
## y_rep[54] 160.13 0.12 5.72 152.86 159.98 167.63 2227 1
## y_rep[55] 149.60 0.12 5.43 142.51 149.54 156.72 2002 1
## y_rep[56] 160.34 0.13 5.55 153.11 160.44 167.64 1969 1
## y_rep[57] 149.53 0.12 5.48 142.56 149.39 156.63 2210 1
## y_rep[58] 160.40 0.12 5.52 153.31 160.43 167.44 2021 1
## y_rep[59] 149.63 0.12 5.50 142.29 149.71 156.61 2144 1
## y_rep[60] 160.47 0.13 5.62 153.28 160.49 167.50 1858 1
## y_rep[61] 149.72 0.13 5.71 142.34 149.82 156.86 1969 1
## y_rep[62] 149.60 0.12 5.56 142.49 149.59 156.78 2180 1
## y_rep[63] 160.31 0.12 5.62 153.16 160.43 167.26 2129 1
## y_rep[64] 160.30 0.12 5.47 153.40 160.16 167.42 2240 1
## y_rep[65] 149.59 0.13 5.72 142.13 149.70 156.88 1925 1
## y_rep[66] 160.24 0.13 5.55 153.35 160.36 167.27 1867 1
## y_rep[67] 149.56 0.13 5.64 142.19 149.42 156.84 1986 1
## y_rep[68] 149.55 0.12 5.47 142.68 149.50 156.65 1995 1
## y_rep[69] 160.47 0.15 5.49 153.39 160.51 167.32 1392 1
## y_rep[70] 149.56 0.13 5.67 142.38 149.53 156.87 2041 1
## y_rep[71] 160.26 0.13 5.58 153.02 160.18 167.33 1952 1
## y_rep[72] 160.46 0.13 5.67 153.24 160.38 167.79 1850 1
## y_rep[73] 160.26 0.13 5.50 153.17 160.21 167.25 1797 1
## y_rep[74] 149.49 0.14 5.59 142.29 149.48 156.84 1620 1
## y_rep[75] 149.50 0.12 5.58 142.67 149.37 156.61 2069 1
## y_rep[76] 149.43 0.12 5.52 142.43 149.41 156.35 2103 1
## y_rep[77] 149.69 0.12 5.53 142.61 149.69 156.74 2211 1
## y_rep[78] 160.41 0.12 5.40 153.54 160.37 167.32 1893 1
## y_rep[79] 149.60 0.14 5.57 142.69 149.58 156.75 1685 1
## y_rep[80] 149.65 0.13 5.41 142.95 149.47 156.81 1871 1
## y_rep[81] 160.21 0.13 5.65 152.98 160.27 167.52 1855 1
## y_rep[82] 160.42 0.13 5.61 153.17 160.28 167.54 1935 1
## y_rep[83] 149.33 0.12 5.43 142.22 149.41 156.38 1920 1
## y_rep[84] 160.28 0.13 5.60 153.16 160.20 167.38 1998 1
## y_rep[85] 149.62 0.14 5.43 142.91 149.67 156.56 1523 1
## y_rep[86] 160.49 0.13 5.61 153.41 160.55 167.89 1982 1
## y_rep[87] 149.68 0.14 5.71 142.33 149.66 157.10 1714 1
## y_rep[88] 160.38 0.13 5.59 153.19 160.27 167.60 1967 1
## y_rep[89] 149.38 0.13 5.64 142.20 149.36 156.54 1963 1
## y_rep[90] 160.27 0.12 5.58 153.30 160.07 167.65 2092 1
## y_rep[91] 160.33 0.12 5.50 153.46 160.41 167.39 2094 1
## y_rep[92] 160.49 0.12 5.40 153.56 160.47 167.36 1978 1
## y_rep[93] 149.61 0.13 5.53 142.57 149.57 156.82 1949 1
## y_rep[94] 149.48 0.12 5.55 142.27 149.63 156.61 1991 1
## y_rep[95] 160.28 0.13 5.56 153.40 160.37 167.24 1961 1
## y_rep[96] 160.27 0.13 5.58 153.10 160.26 167.15 1755 1
## y_rep[97] 149.49 0.13 5.55 142.15 149.65 156.68 1800 1
## y_rep[98] 160.56 0.13 5.59 153.48 160.62 167.77 1958 1
## y_rep[99] 160.19 0.12 5.45 153.38 160.19 167.19 1966 1
## y_rep[100] 149.51 0.12 5.44 142.69 149.36 156.43 1986 1
## y_rep[101] 160.30 0.13 5.48 153.50 160.27 167.33 1915 1
## y_rep[102] 149.58 0.13 5.54 142.66 149.41 156.97 1927 1
## y_rep[103] 149.70 0.12 5.37 142.99 149.58 156.59 2012 1
## y_rep[104] 160.35 0.12 5.67 152.91 160.41 167.49 2074 1
## y_rep[105] 160.30 0.13 5.62 153.15 160.40 167.29 1932 1
## y_rep[106] 149.47 0.13 5.55 142.46 149.44 156.74 1853 1
## y_rep[107] 149.56 0.13 5.61 142.07 149.65 156.62 1989 1
## y_rep[108] 149.51 0.13 5.65 142.26 149.64 156.76 2022 1
## y_rep[109] 160.32 0.12 5.56 153.19 160.22 167.50 1983 1
## y_rep[110] 149.60 0.13 5.65 142.38 149.57 156.70 1958 1
## y_rep[111] 160.20 0.12 5.51 153.04 160.31 167.33 1993 1
## y_rep[112] 149.62 0.13 5.60 142.29 149.72 156.69 1956 1
## y_rep[113] 149.49 0.12 5.69 142.16 149.49 156.93 2073 1
## y_rep[114] 160.25 0.12 5.52 153.19 160.16 167.36 2014 1
## y_rep[115] 160.35 0.13 5.64 153.34 160.38 167.42 1957 1
## y_rep[116] 149.54 0.12 5.53 142.46 149.50 156.64 1962 1
## y_rep[117] 160.27 0.14 5.67 153.04 160.26 167.77 1662 1
## y_rep[118] 160.31 0.13 5.57 153.04 160.52 167.15 1821 1
## y_rep[119] 149.74 0.13 5.56 142.81 149.77 156.75 1827 1
## y_rep[120] 149.62 0.13 5.64 142.38 149.67 156.99 2017 1
## y_rep[121] 160.21 0.13 5.67 152.95 160.27 167.31 1989 1
## y_rep[122] 149.51 0.12 5.40 142.55 149.41 156.49 1920 1
## y_rep[123] 160.60 0.13 5.56 153.56 160.57 167.66 1818 1
## y_rep[124] 149.53 0.12 5.47 142.64 149.27 156.68 1939 1
## y_rep[125] 149.62 0.13 5.64 142.24 149.74 156.56 2011 1
## y_rep[126] 149.51 0.13 5.62 142.43 149.45 156.74 1986 1
## y_rep[127] 149.51 0.13 5.76 142.06 149.48 156.97 1878 1
## y_rep[128] 160.15 0.13 5.59 153.13 160.19 167.34 1795 1
## y_rep[129] 149.54 0.12 5.40 142.71 149.43 156.50 1929 1
## y_rep[130] 149.41 0.12 5.49 142.34 149.29 156.51 1952 1
## y_rep[131] 160.56 0.14 5.63 153.17 160.65 167.95 1719 1
## y_rep[132] 160.23 0.13 5.49 153.05 160.37 167.23 1849 1
## y_rep[133] 149.50 0.13 5.63 142.37 149.42 156.71 2019 1
## y_rep[134] 160.37 0.12 5.59 153.32 160.37 167.33 2008 1
## y_rep[135] 149.51 0.13 5.66 142.31 149.41 156.71 2009 1
## y_rep[136] 160.04 0.13 5.75 152.61 159.99 167.54 2001 1
## y_rep[137] 160.29 0.14 5.66 153.28 160.07 167.68 1708 1
## y_rep[138] 160.20 0.13 5.60 153.06 160.09 167.44 1929 1
## y_rep[139] 149.55 0.12 5.51 142.45 149.61 156.57 2040 1
## y_rep[140] 160.54 0.13 5.55 153.42 160.41 167.68 1868 1
## y_rep[141] 149.65 0.13 5.57 142.53 149.68 156.65 1803 1
## y_rep[142] 149.58 0.13 5.54 142.40 149.64 156.77 1863 1
## y_rep[143] 149.56 0.13 5.64 142.38 149.65 156.70 1877 1
## y_rep[144] 149.61 0.12 5.57 142.53 149.47 156.85 2104 1
## y_rep[145] 149.53 0.12 5.49 142.51 149.51 156.74 2035 1
## y_rep[146] 160.03 0.13 5.54 152.92 160.02 167.24 1930 1
## y_rep[147] 149.68 0.13 5.57 142.68 149.66 156.94 1921 1
## y_rep[148] 149.54 0.13 5.51 142.72 149.45 156.66 1919 1
## y_rep[149] 160.17 0.12 5.42 153.31 160.13 166.93 2060 1
## y_rep[150] 149.53 0.13 5.70 142.41 149.53 156.95 2020 1
## y_rep[151] 160.42 0.13 5.59 153.52 160.35 167.54 1862 1
## y_rep[152] 149.52 0.13 5.50 142.28 149.57 156.47 1908 1
## y_rep[153] 160.16 0.13 5.66 152.99 160.29 167.34 1770 1
## y_rep[154] 149.68 0.13 5.60 142.37 149.72 156.75 1945 1
## y_rep[155] 160.45 0.13 5.53 153.34 160.50 167.39 1950 1
## y_rep[156] 149.55 0.13 5.73 142.00 149.54 157.01 1884 1
## y_rep[157] 149.38 0.12 5.42 142.39 149.43 156.40 2023 1
## y_rep[158] 160.12 0.13 5.57 153.24 160.00 167.14 1955 1
## y_rep[159] 160.34 0.12 5.58 153.21 160.38 167.54 2051 1
## y_rep[160] 149.57 0.13 5.62 142.34 149.73 157.00 1832 1
## y_rep[161] 160.21 0.13 5.61 153.21 160.24 167.25 1970 1
## y_rep[162] 149.43 0.13 5.54 141.98 149.48 156.43 1875 1
## y_rep[163] 160.45 0.13 5.56 153.18 160.40 167.76 1824 1
## y_rep[164] 160.31 0.13 5.58 153.05 160.26 167.68 1990 1
## y_rep[165] 160.34 0.13 5.67 153.11 160.39 167.71 1959 1
## y_rep[166] 149.52 0.13 5.57 142.29 149.48 156.43 1950 1
## y_rep[167] 160.25 0.13 5.61 153.27 160.25 167.33 1871 1
## y_rep[168] 149.68 0.12 5.55 142.89 149.55 156.83 2049 1
## y_rep[169] 149.49 0.13 5.47 142.42 149.51 156.45 1864 1
## y_rep[170] 149.66 0.12 5.54 142.37 149.77 156.64 2190 1
## y_rep[171] 149.62 0.12 5.39 142.63 149.52 156.64 1905 1
## y_rep[172] 160.54 0.14 5.65 153.15 160.53 167.81 1718 1
## y_rep[173] 149.76 0.12 5.48 142.50 149.68 156.83 1979 1
## y_rep[174] 160.44 0.13 5.54 153.49 160.45 167.56 1802 1
## y_rep[175] 149.61 0.12 5.56 142.67 149.53 156.74 2060 1
## y_rep[176] 160.22 0.12 5.66 153.17 160.03 167.42 2135 1
## y_rep[177] 149.67 0.13 5.61 142.38 149.53 156.90 2011 1
## y_rep[178] 149.63 0.12 5.46 142.57 149.70 156.51 1921 1
## y_rep[179] 160.14 0.13 5.60 153.10 160.11 167.33 1986 1
## y_rep[180] 149.66 0.12 5.51 142.62 149.70 156.49 2119 1
## y_rep[181] 159.97 0.12 5.43 153.19 159.85 166.96 1910 1
## y_rep[182] 149.59 0.12 5.53 142.35 149.58 156.77 2118 1
## y_rep[183] 149.68 0.12 5.39 142.86 149.59 156.92 1878 1
## y_rep[184] 149.40 0.12 5.46 142.25 149.38 156.25 2180 1
## y_rep[185] 160.34 0.12 5.52 153.46 160.28 167.45 2015 1
## y_rep[186] 160.10 0.12 5.67 152.97 159.95 167.44 2055 1
## y_rep[187] 149.58 0.12 5.55 142.53 149.62 156.68 2156 1
## y_rep[188] 160.42 0.12 5.59 153.35 160.37 167.54 2019 1
## y_rep[189] 149.46 0.13 5.63 142.38 149.51 156.52 1872 1
## y_rep[190] 160.27 0.13 5.75 152.87 160.28 167.93 1918 1
## y_rep[191] 149.52 0.13 5.47 142.37 149.61 156.28 1863 1
## y_rep[192] 160.43 0.12 5.58 153.43 160.31 167.67 2031 1
## y_rep[193] 149.57 0.13 5.76 142.32 149.48 156.86 1994 1
## y_rep[194] 160.36 0.13 5.64 153.15 160.34 167.44 1937 1
## y_rep[195] 149.41 0.13 5.64 142.38 149.47 156.68 1881 1
## y_rep[196] 149.53 0.13 5.67 142.26 149.46 156.78 1977 1
## y_rep[197] 149.52 0.12 5.58 142.53 149.34 156.93 2009 1
## y_rep[198] 160.39 0.12 5.53 153.43 160.52 167.58 1961 1
## y_rep[199] 160.41 0.14 5.57 153.33 160.45 167.72 1672 1
## y_rep[200] 149.46 0.13 5.55 142.33 149.32 156.43 1959 1
## y_rep[201] 160.26 0.13 5.54 153.21 160.23 167.48 1767 1
## y_rep[202] 149.47 0.13 5.57 142.71 149.35 156.52 1969 1
## y_rep[203] 160.38 0.13 5.66 153.00 160.39 167.74 1882 1
## y_rep[204] 149.65 0.12 5.39 142.72 149.48 156.58 2011 1
## y_rep[205] 160.04 0.13 5.59 152.99 160.03 167.17 1959 1
## y_rep[206] 149.67 0.14 5.59 142.60 149.82 156.66 1690 1
## y_rep[207] 149.61 0.13 5.61 142.45 149.48 156.63 1846 1
## y_rep[208] 149.53 0.13 5.61 142.44 149.43 156.76 1881 1
## y_rep[209] 149.46 0.13 5.60 142.31 149.38 156.62 1938 1
## y_rep[210] 149.68 0.12 5.56 142.53 149.61 156.89 1991 1
## y_rep[211] 160.49 0.13 5.60 153.13 160.59 167.31 1785 1
## y_rep[212] 160.41 0.13 5.57 153.33 160.41 167.38 1961 1
## y_rep[213] 160.46 0.12 5.62 153.32 160.47 167.81 2078 1
## y_rep[214] 149.71 0.13 5.59 142.57 149.79 156.85 1986 1
## y_rep[215] 160.50 0.12 5.51 153.65 160.55 167.68 1984 1
## y_rep[216] 160.64 0.13 5.44 153.62 160.70 167.60 1720 1
## y_rep[217] 160.36 0.12 5.49 153.33 160.38 167.36 1974 1
## y_rep[218] 149.29 0.12 5.52 142.22 149.37 156.46 2036 1
## y_rep[219] 160.23 0.12 5.55 153.09 160.26 167.60 2015 1
## y_rep[220] 149.44 0.12 5.54 142.28 149.50 156.55 2132 1
## y_rep[221] 149.84 0.13 5.59 142.64 149.81 156.98 1985 1
## y_rep[222] 149.67 0.12 5.43 142.77 149.75 156.41 1913 1
## y_rep[223] 160.26 0.13 5.59 152.90 160.24 167.45 1795 1
## y_rep[224] 149.51 0.12 5.46 142.19 149.64 156.45 2041 1
## y_rep[225] 160.20 0.12 5.65 153.11 160.12 167.39 2062 1
## y_rep[226] 149.56 0.12 5.62 142.23 149.54 156.80 2066 1
## y_rep[227] 149.66 0.13 5.74 142.25 149.62 157.20 2023 1
## y_rep[228] 160.39 0.13 5.65 153.20 160.31 167.65 1994 1
## y_rep[229] 149.43 0.12 5.48 142.53 149.47 156.40 2059 1
## y_rep[230] 160.28 0.13 5.52 153.25 160.24 167.19 1822 1
## y_rep[231] 160.32 0.12 5.42 153.37 160.28 167.11 1964 1
## y_rep[232] 149.49 0.13 5.51 142.31 149.53 156.53 1897 1
## y_rep[233] 149.61 0.12 5.56 142.40 149.52 156.49 2011 1
## y_rep[234] 160.37 0.12 5.46 153.44 160.41 167.39 1941 1
## y_rep[235] 160.22 0.12 5.60 152.94 160.22 167.46 2010 1
## y_rep[236] 149.49 0.12 5.43 142.63 149.42 156.46 1894 1
## y_rep[237] 160.35 0.12 5.66 153.16 160.52 167.34 2121 1
## y_rep[238] 149.54 0.12 5.70 142.49 149.52 156.78 2114 1
## y_rep[239] 149.70 0.13 5.51 142.76 149.66 156.74 1914 1
## y_rep[240] 149.45 0.12 5.52 142.33 149.55 156.37 2064 1
## y_rep[241] 149.37 0.12 5.54 142.26 149.47 156.54 2067 1
## y_rep[242] 160.35 0.13 5.54 153.33 160.38 167.48 1804 1
## y_rep[243] 160.30 0.13 5.68 153.07 160.21 167.63 1991 1
## y_rep[244] 149.47 0.13 5.61 142.34 149.46 156.72 1897 1
## y_rep[245] 160.29 0.13 5.49 153.24 160.34 167.31 1886 1
## y_rep[246] 149.43 0.13 5.50 142.28 149.38 156.75 1811 1
## y_rep[247] 160.31 0.13 5.57 153.15 160.25 167.48 1893 1
## y_rep[248] 149.63 0.13 5.58 142.70 149.62 156.48 1881 1
## y_rep[249] 160.32 0.12 5.52 153.32 160.40 167.34 2124 1
## y_rep[250] 160.45 0.13 5.58 153.29 160.40 167.62 1894 1
## y_rep[251] 160.31 0.12 5.60 153.07 160.32 167.61 2086 1
## y_rep[252] 160.31 0.13 5.40 153.37 160.23 166.85 1863 1
## y_rep[253] 149.57 0.13 5.65 142.22 149.50 156.82 1919 1
## y_rep[254] 149.55 0.13 5.54 142.52 149.56 156.46 1903 1
## y_rep[255] 149.46 0.12 5.47 142.45 149.31 156.34 1968 1
## y_rep[256] 149.64 0.13 5.56 142.75 149.51 156.80 1802 1
## y_rep[257] 160.17 0.12 5.44 153.17 160.17 167.22 1970 1
## y_rep[258] 149.68 0.12 5.56 142.52 149.60 156.71 2073 1
## y_rep[259] 160.37 0.12 5.55 153.03 160.45 167.69 2011 1
## y_rep[260] 160.32 0.13 5.43 153.48 160.45 167.20 1882 1
## y_rep[261] 160.44 0.13 5.65 153.02 160.52 167.80 1977 1
## y_rep[262] 149.58 0.12 5.49 142.58 149.53 156.77 2001 1
## y_rep[263] 149.56 0.14 5.56 142.32 149.73 156.40 1646 1
## y_rep[264] 149.52 0.13 5.64 142.21 149.60 156.64 1860 1
## y_rep[265] 160.45 0.13 5.57 153.37 160.44 167.49 1908 1
## y_rep[266] 149.55 0.12 5.57 142.31 149.54 156.57 2128 1
## y_rep[267] 160.55 0.13 5.64 153.17 160.61 167.77 1921 1
## y_rep[268] 149.57 0.12 5.49 142.72 149.57 156.62 2050 1
## y_rep[269] 149.62 0.14 5.73 142.34 149.51 156.88 1763 1
## y_rep[270] 160.34 0.13 5.52 153.31 160.16 167.50 1920 1
## y_rep[271] 160.36 0.13 5.55 153.19 160.40 167.27 1828 1
## y_rep[272] 149.45 0.13 5.61 142.31 149.35 156.77 1909 1
## y_rep[273] 160.18 0.12 5.56 153.26 160.33 167.25 2034 1
## y_rep[274] 160.43 0.12 5.55 153.17 160.51 167.35 1998 1
## y_rep[275] 149.83 0.12 5.59 142.77 149.74 156.90 2161 1
## y_rep[276] 160.38 0.12 5.47 153.27 160.49 167.20 2199 1
## y_rep[277] 149.68 0.13 5.52 142.69 149.76 156.77 1858 1
## y_rep[278] 149.29 0.13 5.55 142.20 149.43 156.31 1801 1
## y_rep[279] 160.07 0.12 5.45 153.07 160.08 166.99 1958 1
## y_rep[280] 149.51 0.13 5.52 142.48 149.44 156.54 1897 1
## y_rep[281] 149.49 0.12 5.53 142.22 149.52 156.55 2072 1
## y_rep[282] 160.56 0.13 5.76 153.26 160.56 168.04 1916 1
## y_rep[283] 149.73 0.13 5.49 142.63 149.84 156.76 1877 1
## y_rep[284] 160.16 0.13 5.48 153.18 160.02 167.19 1702 1
## y_rep[285] 149.76 0.13 5.43 142.75 149.79 156.72 1684 1
## y_rep[286] 160.24 0.13 5.47 153.25 160.48 167.06 1898 1
## y_rep[287] 160.32 0.12 5.63 153.22 160.36 167.55 2133 1
## y_rep[288] 160.26 0.13 5.61 153.17 160.36 167.15 1932 1
## y_rep[289] 149.39 0.12 5.41 142.53 149.30 156.58 1923 1
## y_rep[290] 160.23 0.13 5.50 153.12 160.15 167.17 1850 1
## y_rep[291] 160.26 0.11 5.46 153.21 160.35 167.24 2354 1
## y_rep[292] 160.43 0.12 5.59 153.22 160.42 167.54 2014 1
## y_rep[293] 160.11 0.12 5.55 152.91 160.05 167.22 2217 1
## y_rep[294] 149.30 0.13 5.66 142.13 149.34 156.49 1853 1
## y_rep[295] 160.28 0.12 5.60 153.31 160.21 167.50 2137 1
## y_rep[296] 149.64 0.12 5.52 142.54 149.53 156.85 2123 1
## y_rep[297] 149.51 0.12 5.65 142.42 149.49 156.75 2058 1
## y_rep[298] 149.45 0.12 5.39 142.58 149.50 156.19 1922 1
## y_rep[299] 160.47 0.12 5.39 153.55 160.51 167.65 1958 1
## y_rep[300] 160.41 0.13 5.58 153.38 160.50 167.84 1740 1
## y_rep[301] 149.82 0.13 5.47 142.74 149.83 156.87 1894 1
## y_rep[302] 160.24 0.12 5.56 153.26 160.13 167.07 2182 1
## y_rep[303] 149.55 0.13 5.45 142.61 149.57 156.55 1851 1
## y_rep[304] 160.29 0.12 5.45 153.15 160.21 167.45 2010 1
## y_rep[305] 160.21 0.13 5.52 153.27 160.16 167.29 1839 1
## y_rep[306] 160.32 0.13 5.65 153.18 160.33 167.41 1844 1
## y_rep[307] 160.20 0.13 5.60 152.97 160.07 167.37 1818 1
## y_rep[308] 149.65 0.13 5.64 142.54 149.64 156.78 1946 1
## y_rep[309] 149.40 0.12 5.57 142.64 149.32 156.52 2113 1
## y_rep[310] 149.55 0.13 5.55 142.37 149.71 156.27 1957 1
## y_rep[311] 149.41 0.12 5.60 142.45 149.35 156.54 2020 1
## y_rep[312] 149.46 0.13 5.44 142.40 149.53 156.34 1735 1
## y_rep[313] 160.40 0.12 5.63 153.30 160.41 167.79 2254 1
## y_rep[314] 149.60 0.13 5.53 142.61 149.69 156.48 1899 1
## y_rep[315] 160.52 0.13 5.63 153.17 160.56 167.96 2024 1
## y_rep[316] 149.59 0.13 5.64 142.56 149.63 156.76 2003 1
## y_rep[317] 160.40 0.12 5.52 153.35 160.45 167.52 2013 1
## y_rep[318] 149.61 0.12 5.58 142.60 149.51 156.79 2008 1
## y_rep[319] 160.32 0.12 5.64 153.10 160.21 167.60 2178 1
## y_rep[320] 149.61 0.13 5.54 142.50 149.74 156.49 1959 1
## y_rep[321] 160.25 0.14 5.62 153.08 160.29 167.35 1508 1
## y_rep[322] 149.35 0.12 5.54 142.38 149.41 156.50 2120 1
## y_rep[323] 160.34 0.12 5.45 152.85 160.42 167.21 2009 1
## y_rep[324] 160.37 0.12 5.49 153.38 160.39 167.46 1998 1
## y_rep[325] 149.54 0.13 5.67 142.30 149.45 156.93 1950 1
## y_rep[326] 149.47 0.12 5.52 142.38 149.44 156.36 2017 1
## y_rep[327] 149.69 0.12 5.47 142.78 149.73 156.69 1958 1
## y_rep[328] 160.40 0.13 5.63 153.06 160.48 167.50 1846 1
## y_rep[329] 160.36 0.13 5.70 153.02 160.47 167.55 1927 1
## y_rep[330] 160.25 0.14 5.68 153.12 160.30 167.67 1748 1
## y_rep[331] 160.17 0.12 5.52 152.98 160.16 166.99 2115 1
## y_rep[332] 159.95 0.13 5.49 152.56 159.98 166.96 1878 1
## y_rep[333] 149.59 0.12 5.52 142.73 149.51 156.59 1995 1
## y_rep[334] 160.47 0.12 5.54 153.34 160.58 167.28 2027 1
## y_rep[335] 149.45 0.13 5.75 141.82 149.49 156.96 1962 1
## y_rep[336] 160.15 0.12 5.49 153.06 160.19 167.13 2158 1
## y_rep[337] 149.50 0.13 5.64 142.01 149.63 156.61 1944 1
## y_rep[338] 149.55 0.12 5.60 142.19 149.50 156.61 2034 1
## y_rep[339] 149.53 0.12 5.69 142.30 149.49 156.83 2216 1
## y_rep[340] 149.38 0.13 5.70 142.14 149.38 156.77 1841 1
## y_rep[341] 149.58 0.12 5.49 142.73 149.64 156.64 2087 1
## y_rep[342] 160.07 0.13 5.62 152.81 160.10 167.15 1908 1
## y_rep[343] 149.35 0.13 5.51 142.24 149.37 156.32 1792 1
## y_rep[344] 160.18 0.12 5.66 152.80 160.15 167.24 2055 1
## y_rep[345] 149.43 0.14 5.70 142.28 149.35 156.56 1765 1
## y_rep[346] 160.48 0.13 5.67 153.31 160.50 167.72 1953 1
## y_rep[347] 149.67 0.13 5.53 142.60 149.71 156.75 1760 1
## y_rep[348] 149.65 0.13 5.56 142.51 149.82 156.58 1889 1
## y_rep[349] 149.43 0.12 5.56 142.53 149.33 156.46 2048 1
## y_rep[350] 160.38 0.12 5.50 153.43 160.34 167.51 1992 1
## y_rep[351] 149.44 0.13 5.50 142.36 149.40 156.42 1848 1
## y_rep[352] 160.34 0.13 5.49 153.33 160.26 167.39 1849 1
## lp__ -781.30 0.04 1.19 -782.90 -780.98 -780.12 743 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:54:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).4.2 Index variable
Here we consider the variable male as an index variable, with value 1 whenever the person is a female and a value 2 whenever the person is a male.
howell2$male.index <- ifelse(howell2$male==1,2,1)
m5.2.data <- list(N = nrow(howell2), height = howell2$height, male = as.integer(howell2$male.index), n = length(unique(howell2$male.index)))
str(m5.2.data)## List of 4
## $ N : int 352
## $ height: num [1:352] 152 140 137 157 145 ...
## $ male : int [1:352] 2 1 1 2 1 2 1 2 1 2 ...
## $ n : int 2Model equation:
\[\begin{align*} h_{i} &\sim \mathcal{N}(\mu_{i} \, , \, \sigma^2) \\[4pt] \mu_{i} &= \alpha_{\text{SEX}[\textit{i}]}\\[4pt] \alpha_{j} &\sim \mathcal{N}(178,20) \quad \text{, for} \, j = 1..2\\[4pt] \sigma^2 &\sim \mathcal{U}(0,5)\\[4pt] \end{align*}\]
data {
int<lower=0> N; // number of observations
int<lower=0> n; // number of levels of the categorical variable
vector[N] height; // response variables
int<lower=1,upper=n> male[N]; // categorical variable
}
parameters {
vector[n] beta; // real beta[n]; (idem) // intercepts of the categorical variable
real sigma;
}
transformed parameters {
vector[N] mu; // create one variable with predictions for each obs
for (i in 1:N) // loop for all cases
mu[i] = beta[male[i]];
}
model {
height ~ normal(mu ,sigma); // likelihood
sigma ~ cauchy(0,5);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2 <- sampling(m5.2, data = m5.2.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 0da8a26b283bfab1846d7d15060893cd.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## beta[2] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## sigma 5.55 0.00 0.20 5.29 5.54 5.81 1800 1
## mu[1] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[2] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[3] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[4] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[5] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[6] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[7] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[8] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[9] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[10] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[11] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[12] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[13] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[14] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[15] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[16] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[17] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[18] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[19] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[20] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[21] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[22] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[23] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[24] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[25] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[26] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[27] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[28] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[29] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[30] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[31] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[32] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[33] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[34] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[35] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[36] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[37] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[38] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[39] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[40] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[41] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[42] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[43] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[44] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[45] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[46] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[47] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[48] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[49] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[50] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[51] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[52] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[53] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[54] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[55] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[56] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[57] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[58] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[59] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[60] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[61] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[62] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[63] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[64] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[65] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[66] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[67] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[68] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[69] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[70] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[71] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[72] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[73] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[74] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[75] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[76] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[77] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[78] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[79] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[80] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[81] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[82] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[83] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[84] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[85] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[86] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[87] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[88] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[89] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[90] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[91] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[92] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[93] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[94] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[95] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[96] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[97] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[98] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[99] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[100] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[101] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[102] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[103] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[104] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[105] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[106] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[107] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[108] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[109] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[110] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[111] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[112] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[113] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[114] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[115] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[116] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[117] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[118] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[119] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[120] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[121] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[122] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[123] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[124] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[125] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[126] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[127] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[128] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[129] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[130] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[131] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[132] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[133] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[134] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[135] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[136] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[137] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[138] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[139] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[140] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[141] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[142] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[143] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[144] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[145] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[146] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[147] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[148] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[149] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[150] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[151] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[152] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[153] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[154] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[155] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[156] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[157] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[158] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[159] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[160] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[161] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[162] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[163] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[164] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[165] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[166] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[167] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[168] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[169] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[170] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[171] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[172] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[173] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[174] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[175] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[176] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[177] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[178] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[179] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[180] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[181] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[182] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[183] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[184] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[185] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[186] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[187] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[188] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[189] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[190] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[191] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[192] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[193] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[194] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[195] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[196] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[197] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[198] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[199] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[200] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[201] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[202] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[203] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[204] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[205] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[206] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[207] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[208] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[209] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[210] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[211] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[212] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[213] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[214] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[215] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[216] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[217] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[218] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[219] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[220] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[221] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[222] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[223] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[224] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[225] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[226] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[227] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[228] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[229] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[230] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[231] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[232] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[233] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[234] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[235] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[236] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[237] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[238] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[239] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[240] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[241] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[242] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[243] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[244] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[245] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[246] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[247] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[248] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[249] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[250] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[251] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[252] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[253] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[254] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[255] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[256] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[257] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[258] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[259] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[260] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[261] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[262] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[263] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[264] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[265] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[266] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[267] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[268] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[269] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[270] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[271] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[272] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[273] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[274] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[275] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[276] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[277] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[278] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[279] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[280] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[281] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[282] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[283] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[284] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[285] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[286] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[287] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[288] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[289] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[290] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[291] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[292] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[293] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[294] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[295] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[296] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[297] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[298] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[299] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[300] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[301] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[302] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[303] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[304] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[305] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[306] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[307] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[308] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[309] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[310] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[311] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[312] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[313] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[314] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[315] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[316] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[317] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[318] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[319] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[320] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[321] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[322] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[323] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[324] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[325] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[326] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[327] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[328] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[329] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[330] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[331] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[332] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[333] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[334] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[335] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[336] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[337] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[338] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[339] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[340] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[341] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[342] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[343] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[344] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[345] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[346] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[347] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[348] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[349] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[350] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## mu[351] 149.51 0.01 0.40 148.99 149.51 150.01 1825 1
## mu[352] 160.36 0.01 0.46 159.77 160.37 160.93 1728 1
## y_rep[1] 160.40 0.12 5.61 153.41 160.36 167.66 2095 1
## y_rep[2] 149.15 0.13 5.57 142.02 149.10 156.26 1974 1
## y_rep[3] 149.55 0.12 5.47 142.54 149.56 156.49 2057 1
## y_rep[4] 160.35 0.13 5.60 153.11 160.43 167.58 1911 1
## y_rep[5] 149.36 0.12 5.50 142.32 149.32 156.38 2074 1
## y_rep[6] 160.35 0.12 5.57 153.14 160.30 167.29 2285 1
## y_rep[7] 149.65 0.13 5.61 142.28 149.63 156.83 1978 1
## y_rep[8] 160.25 0.13 5.61 153.07 160.30 167.16 1974 1
## y_rep[9] 149.61 0.13 5.71 142.30 149.52 157.01 1992 1
## y_rep[10] 160.40 0.12 5.60 153.24 160.27 167.82 2093 1
## y_rep[11] 149.73 0.12 5.42 142.86 149.73 156.75 1962 1
## y_rep[12] 160.36 0.12 5.52 153.38 160.33 167.46 2019 1
## y_rep[13] 149.46 0.12 5.62 142.39 149.39 156.63 2055 1
## y_rep[14] 149.59 0.14 5.69 142.44 149.45 156.67 1764 1
## y_rep[15] 149.61 0.13 5.55 142.43 149.53 156.46 1926 1
## y_rep[16] 160.36 0.13 5.52 153.35 160.37 167.52 1916 1
## y_rep[17] 160.49 0.13 5.59 153.56 160.38 167.66 1902 1
## y_rep[18] 149.65 0.13 5.56 142.41 149.58 156.70 1957 1
## y_rep[19] 160.49 0.13 5.64 153.47 160.44 167.67 1849 1
## y_rep[20] 149.45 0.13 5.63 142.48 149.47 156.57 1946 1
## y_rep[21] 160.43 0.13 5.66 153.49 160.18 167.93 1793 1
## y_rep[22] 149.32 0.13 5.59 142.13 149.45 156.62 1951 1
## y_rep[23] 149.28 0.12 5.50 142.22 149.28 156.15 2055 1
## y_rep[24] 160.50 0.14 5.57 153.31 160.59 167.53 1667 1
## y_rep[25] 160.38 0.13 5.62 153.50 160.33 167.62 1907 1
## y_rep[26] 149.62 0.13 5.63 142.45 149.66 156.79 1967 1
## y_rep[27] 149.49 0.13 5.58 142.45 149.49 156.62 1962 1
## y_rep[28] 149.70 0.12 5.61 142.50 149.69 156.91 2090 1
## y_rep[29] 149.42 0.12 5.42 142.62 149.30 156.55 1908 1
## y_rep[30] 149.47 0.12 5.54 142.18 149.56 156.58 2058 1
## y_rep[31] 149.52 0.12 5.44 142.62 149.49 156.34 1922 1
## y_rep[32] 160.45 0.13 5.65 153.27 160.55 167.82 1885 1
## y_rep[33] 160.32 0.13 5.51 153.31 160.34 167.39 1937 1
## y_rep[34] 160.29 0.13 5.57 153.30 160.33 167.31 1908 1
## y_rep[35] 160.40 0.13 5.68 153.17 160.61 167.71 1929 1
## y_rep[36] 149.68 0.12 5.44 142.67 149.62 156.79 2107 1
## y_rep[37] 149.66 0.13 5.47 142.78 149.71 156.73 1835 1
## y_rep[38] 160.45 0.12 5.57 153.21 160.19 167.72 2015 1
## y_rep[39] 149.36 0.13 5.67 142.17 149.27 156.70 1975 1
## y_rep[40] 160.49 0.14 5.81 153.14 160.32 167.90 1816 1
## y_rep[41] 160.46 0.12 5.49 153.27 160.48 167.50 1947 1
## y_rep[42] 160.45 0.12 5.54 153.34 160.52 167.78 2097 1
## y_rep[43] 149.37 0.12 5.64 142.35 149.34 156.75 2045 1
## y_rep[44] 149.31 0.13 5.66 141.97 149.18 156.46 1928 1
## y_rep[45] 149.33 0.13 5.58 142.12 149.47 156.41 1955 1
## y_rep[46] 149.53 0.13 5.59 142.25 149.60 156.69 1767 1
## y_rep[47] 160.33 0.13 5.66 153.00 160.49 167.50 1911 1
## y_rep[48] 149.56 0.13 5.60 142.27 149.76 156.57 1963 1
## y_rep[49] 160.25 0.12 5.63 152.99 160.22 167.39 2110 1
## y_rep[50] 149.39 0.13 5.67 142.10 149.34 156.61 1929 1
## y_rep[51] 149.37 0.14 5.76 142.07 149.36 156.81 1646 1
## y_rep[52] 149.38 0.13 5.64 142.18 149.38 156.74 1909 1
## y_rep[53] 160.19 0.12 5.45 153.03 160.22 167.35 2037 1
## y_rep[54] 160.21 0.13 5.61 153.09 160.35 167.42 1778 1
## y_rep[55] 149.68 0.13 5.57 142.59 149.84 156.67 1938 1
## y_rep[56] 160.32 0.12 5.43 153.22 160.32 167.23 2032 1
## y_rep[57] 149.52 0.12 5.55 142.26 149.47 156.59 2088 1
## y_rep[58] 160.43 0.13 5.51 153.27 160.52 167.74 1840 1
## y_rep[59] 149.44 0.13 5.55 142.35 149.63 156.54 1889 1
## y_rep[60] 160.31 0.12 5.45 153.57 160.06 167.44 1984 1
## y_rep[61] 149.64 0.13 5.56 142.33 149.73 156.76 1937 1
## y_rep[62] 149.59 0.14 5.72 142.52 149.55 156.98 1727 1
## y_rep[63] 160.50 0.12 5.48 153.57 160.25 167.55 1945 1
## y_rep[64] 160.51 0.13 5.47 153.66 160.47 167.61 1816 1
## y_rep[65] 149.49 0.13 5.56 142.44 149.45 156.53 1839 1
## y_rep[66] 160.36 0.13 5.47 153.23 160.39 167.34 1827 1
## y_rep[67] 149.64 0.13 5.58 142.62 149.58 156.72 1853 1
## y_rep[68] 149.24 0.12 5.60 141.90 149.11 156.35 2082 1
## y_rep[69] 160.22 0.12 5.49 153.46 160.19 167.43 1966 1
## y_rep[70] 149.46 0.13 5.50 142.63 149.41 156.80 1747 1
## y_rep[71] 160.25 0.12 5.51 152.97 160.46 167.23 2229 1
## y_rep[72] 160.24 0.13 5.49 153.18 160.38 167.11 1866 1
## y_rep[73] 160.45 0.13 5.65 153.06 160.43 167.94 1976 1
## y_rep[74] 149.50 0.12 5.54 142.26 149.57 156.73 2027 1
## y_rep[75] 149.49 0.12 5.54 142.55 149.45 156.61 2032 1
## y_rep[76] 149.62 0.13 5.60 142.39 149.63 156.67 1753 1
## y_rep[77] 149.51 0.13 5.61 142.40 149.68 156.84 1939 1
## y_rep[78] 160.09 0.12 5.50 153.06 160.15 167.12 1970 1
## y_rep[79] 149.61 0.12 5.51 142.47 149.79 156.73 1991 1
## y_rep[80] 149.59 0.13 5.51 142.45 149.44 156.69 1885 1
## y_rep[81] 160.41 0.13 5.48 153.49 160.44 167.32 1919 1
## y_rep[82] 160.27 0.12 5.54 153.08 160.15 167.60 2113 1
## y_rep[83] 149.49 0.12 5.53 142.55 149.54 156.78 2044 1
## y_rep[84] 160.09 0.12 5.46 152.99 160.07 167.21 1979 1
## y_rep[85] 149.46 0.13 5.70 142.39 149.41 156.81 1907 1
## y_rep[86] 160.31 0.12 5.62 152.99 160.34 167.48 2089 1
## y_rep[87] 149.38 0.12 5.53 142.19 149.39 156.52 1977 1
## y_rep[88] 160.32 0.12 5.64 153.27 160.34 167.49 2201 1
## y_rep[89] 149.70 0.12 5.54 142.77 149.63 156.72 2042 1
## y_rep[90] 160.41 0.12 5.49 153.36 160.45 167.60 2057 1
## y_rep[91] 160.27 0.12 5.58 153.14 160.31 167.51 2033 1
## y_rep[92] 160.56 0.13 5.61 153.55 160.47 167.73 1928 1
## y_rep[93] 149.57 0.12 5.46 142.75 149.50 156.56 1974 1
## y_rep[94] 149.41 0.13 5.41 142.47 149.35 156.46 1761 1
## y_rep[95] 160.46 0.12 5.61 153.22 160.50 167.75 2070 1
## y_rep[96] 160.44 0.13 5.60 153.28 160.57 167.80 1843 1
## y_rep[97] 149.46 0.12 5.58 142.32 149.52 156.70 2054 1
## y_rep[98] 160.17 0.12 5.64 153.05 160.17 167.54 2175 1
## y_rep[99] 160.38 0.12 5.47 153.49 160.36 167.48 2025 1
## y_rep[100] 149.44 0.13 5.59 142.17 149.59 156.64 1936 1
## y_rep[101] 160.46 0.12 5.45 153.57 160.42 167.68 2118 1
## y_rep[102] 149.52 0.13 5.56 142.57 149.55 156.73 1824 1
## y_rep[103] 149.61 0.13 5.66 142.20 149.72 157.00 1917 1
## y_rep[104] 160.46 0.14 5.68 153.04 160.48 167.74 1688 1
## y_rep[105] 160.39 0.13 5.54 153.48 160.44 167.46 1823 1
## y_rep[106] 149.62 0.13 5.70 142.29 149.64 156.90 2069 1
## y_rep[107] 149.54 0.12 5.63 142.53 149.60 156.85 2090 1
## y_rep[108] 149.63 0.13 5.57 142.47 149.55 156.70 1982 1
## y_rep[109] 160.35 0.12 5.63 153.16 160.21 167.77 2170 1
## y_rep[110] 149.43 0.13 5.73 142.08 149.41 156.69 2051 1
## y_rep[111] 160.42 0.12 5.69 153.25 160.30 167.73 2104 1
## y_rep[112] 149.54 0.12 5.50 142.41 149.60 156.77 1948 1
## y_rep[113] 149.48 0.12 5.49 142.45 149.55 156.64 1990 1
## y_rep[114] 160.16 0.13 5.58 152.97 160.24 167.29 1891 1
## y_rep[115] 160.52 0.12 5.46 153.47 160.38 167.45 2150 1
## y_rep[116] 149.51 0.13 5.60 142.22 149.41 156.80 1738 1
## y_rep[117] 160.31 0.13 5.37 153.49 160.24 167.27 1831 1
## y_rep[118] 160.02 0.13 5.47 152.93 160.12 167.13 1884 1
## y_rep[119] 149.54 0.13 5.51 142.31 149.51 156.55 1867 1
## y_rep[120] 149.60 0.12 5.54 142.56 149.60 156.66 2091 1
## y_rep[121] 160.55 0.12 5.57 153.38 160.51 167.83 2186 1
## y_rep[122] 149.38 0.13 5.58 142.01 149.54 156.56 1881 1
## y_rep[123] 160.37 0.12 5.48 153.48 160.36 167.13 2171 1
## y_rep[124] 149.49 0.13 5.39 142.57 149.58 156.33 1801 1
## y_rep[125] 149.65 0.12 5.60 142.54 149.55 157.16 2043 1
## y_rep[126] 149.28 0.13 5.66 142.11 149.15 156.74 1774 1
## y_rep[127] 149.83 0.13 5.78 142.42 149.86 157.26 2043 1
## y_rep[128] 160.14 0.13 5.66 153.10 160.04 167.44 2030 1
## y_rep[129] 149.40 0.12 5.50 142.28 149.51 156.49 2113 1
## y_rep[130] 149.47 0.12 5.57 142.56 149.35 156.85 2015 1
## y_rep[131] 160.38 0.14 5.57 153.38 160.29 167.44 1626 1
## y_rep[132] 160.37 0.13 5.47 153.27 160.51 167.42 1886 1
## y_rep[133] 149.50 0.13 5.59 142.52 149.38 156.92 1840 1
## y_rep[134] 160.38 0.12 5.51 153.33 160.43 167.58 2113 1
## y_rep[135] 149.31 0.12 5.54 142.19 149.38 156.16 2041 1
## y_rep[136] 160.27 0.12 5.53 153.23 160.13 167.57 2122 1
## y_rep[137] 160.18 0.12 5.57 152.97 160.09 167.19 2222 1
## y_rep[138] 160.28 0.12 5.55 153.26 160.23 167.46 2121 1
## y_rep[139] 149.38 0.13 5.48 142.38 149.25 156.32 1809 1
## y_rep[140] 160.11 0.12 5.52 153.28 160.15 167.29 1989 1
## y_rep[141] 149.63 0.12 5.49 142.68 149.56 156.59 2070 1
## y_rep[142] 149.47 0.13 5.58 142.33 149.51 156.55 1934 1
## y_rep[143] 149.44 0.12 5.40 142.50 149.47 156.36 1946 1
## y_rep[144] 149.49 0.12 5.48 142.38 149.49 156.40 2042 1
## y_rep[145] 149.48 0.13 5.45 142.76 149.40 156.48 1809 1
## y_rep[146] 160.36 0.13 5.57 153.47 160.36 167.40 1842 1
## y_rep[147] 149.69 0.12 5.53 142.58 149.76 156.79 1963 1
## y_rep[148] 149.43 0.14 5.70 142.16 149.46 156.66 1770 1
## y_rep[149] 160.14 0.13 5.59 152.72 160.17 167.25 1819 1
## y_rep[150] 149.26 0.12 5.47 142.21 149.18 156.37 2141 1
## y_rep[151] 160.36 0.13 5.79 152.84 160.44 167.81 1930 1
## y_rep[152] 149.52 0.13 5.66 142.40 149.43 156.81 1849 1
## y_rep[153] 160.39 0.12 5.49 153.51 160.33 167.52 2018 1
## y_rep[154] 149.45 0.13 5.61 142.33 149.50 156.48 1938 1
## y_rep[155] 160.21 0.12 5.58 153.17 160.06 167.49 2083 1
## y_rep[156] 149.29 0.13 5.57 142.02 149.40 156.39 1836 1
## y_rep[157] 149.69 0.12 5.43 142.79 149.74 156.86 1891 1
## y_rep[158] 160.22 0.13 5.71 153.16 160.29 167.55 1950 1
## y_rep[159] 160.46 0.12 5.57 153.19 160.47 167.59 2008 1
## y_rep[160] 149.57 0.13 5.49 142.67 149.45 156.76 1876 1
## y_rep[161] 160.26 0.13 5.67 153.12 160.25 167.47 1996 1
## y_rep[162] 149.64 0.12 5.55 142.75 149.70 156.69 2043 1
## y_rep[163] 160.27 0.13 5.68 152.83 160.16 167.54 1850 1
## y_rep[164] 160.41 0.14 5.76 153.29 160.63 167.59 1788 1
## y_rep[165] 160.42 0.12 5.68 152.95 160.44 167.77 2098 1
## y_rep[166] 149.25 0.13 5.43 142.32 149.27 156.29 1720 1
## y_rep[167] 160.26 0.12 5.53 153.13 160.26 167.29 1973 1
## y_rep[168] 149.64 0.14 5.64 142.43 149.67 156.85 1722 1
## y_rep[169] 149.28 0.13 5.61 142.20 149.34 156.40 1974 1
## y_rep[170] 149.48 0.13 5.64 142.42 149.42 156.85 1885 1
## y_rep[171] 149.31 0.13 5.48 142.25 149.18 156.56 1672 1
## y_rep[172] 160.43 0.12 5.42 153.39 160.50 167.28 1984 1
## y_rep[173] 149.25 0.13 5.69 142.02 149.34 156.64 1967 1
## y_rep[174] 160.60 0.12 5.60 153.64 160.56 167.92 2053 1
## y_rep[175] 149.67 0.13 5.62 142.67 149.72 156.80 1870 1
## y_rep[176] 160.13 0.12 5.46 153.14 160.23 166.87 2040 1
## y_rep[177] 149.63 0.14 5.55 142.50 149.71 156.45 1634 1
## y_rep[178] 149.53 0.12 5.67 142.37 149.42 156.81 2172 1
## y_rep[179] 160.21 0.13 5.62 152.87 160.28 167.38 2012 1
## y_rep[180] 149.30 0.12 5.36 142.46 149.31 156.12 2041 1
## y_rep[181] 160.22 0.12 5.54 153.34 160.24 167.20 2034 1
## y_rep[182] 149.34 0.12 5.39 142.23 149.41 156.19 2009 1
## y_rep[183] 149.51 0.13 5.67 142.20 149.47 156.95 2029 1
## y_rep[184] 149.56 0.12 5.43 142.57 149.66 156.34 1930 1
## y_rep[185] 160.37 0.13 5.55 153.20 160.56 167.36 1919 1
## y_rep[186] 160.47 0.12 5.67 153.07 160.47 167.55 2109 1
## y_rep[187] 149.49 0.13 5.62 142.29 149.57 156.70 1776 1
## y_rep[188] 160.42 0.12 5.47 153.30 160.51 167.35 1963 1
## y_rep[189] 149.50 0.13 5.60 142.34 149.39 156.79 1939 1
## y_rep[190] 160.39 0.12 5.63 153.16 160.41 167.40 2075 1
## y_rep[191] 149.55 0.12 5.49 142.62 149.72 156.53 2051 1
## y_rep[192] 160.50 0.13 5.53 153.40 160.51 167.64 1897 1
## y_rep[193] 149.46 0.13 5.68 142.24 149.41 156.99 1969 1
## y_rep[194] 160.33 0.13 5.69 152.90 160.32 167.52 1992 1
## y_rep[195] 149.61 0.13 5.65 142.44 149.65 156.69 1898 1
## y_rep[196] 149.51 0.13 5.62 142.12 149.61 156.48 1793 1
## y_rep[197] 149.51 0.12 5.48 142.35 149.53 156.38 2059 1
## y_rep[198] 160.50 0.13 5.67 153.23 160.44 167.84 2012 1
## y_rep[199] 160.43 0.13 5.63 153.38 160.24 167.75 1832 1
## y_rep[200] 149.31 0.12 5.66 142.34 149.20 156.60 2075 1
## y_rep[201] 160.27 0.13 5.61 153.11 160.24 167.50 1967 1
## y_rep[202] 149.46 0.12 5.63 142.43 149.52 156.46 2146 1
## y_rep[203] 160.42 0.12 5.60 153.34 160.34 167.69 2214 1
## y_rep[204] 149.49 0.13 5.70 142.16 149.39 156.85 1951 1
## y_rep[205] 160.28 0.12 5.41 153.24 160.27 167.09 2019 1
## y_rep[206] 149.64 0.13 5.47 142.83 149.56 156.79 1895 1
## y_rep[207] 149.59 0.12 5.63 142.41 149.48 156.74 2032 1
## y_rep[208] 149.44 0.14 5.51 142.53 149.29 156.62 1609 1
## y_rep[209] 149.59 0.12 5.55 142.65 149.55 156.75 2109 1
## y_rep[210] 149.57 0.13 5.70 142.44 149.55 157.00 1928 1
## y_rep[211] 160.42 0.12 5.55 153.34 160.27 167.59 2065 1
## y_rep[212] 160.50 0.12 5.53 153.56 160.77 167.65 2114 1
## y_rep[213] 160.50 0.12 5.59 153.25 160.48 167.52 2033 1
## y_rep[214] 149.59 0.12 5.67 142.33 149.78 156.63 2066 1
## y_rep[215] 160.19 0.12 5.54 153.32 160.05 167.32 2037 1
## y_rep[216] 160.32 0.12 5.52 153.39 160.35 167.22 1962 1
## y_rep[217] 160.38 0.13 5.61 153.13 160.38 167.73 1972 1
## y_rep[218] 149.49 0.13 5.60 142.30 149.54 156.77 1970 1
## y_rep[219] 160.36 0.12 5.59 153.18 160.39 167.57 2285 1
## y_rep[220] 149.69 0.12 5.67 142.55 149.75 156.65 2061 1
## y_rep[221] 149.44 0.14 5.76 141.92 149.31 157.05 1814 1
## y_rep[222] 149.46 0.13 5.56 142.24 149.37 156.44 1947 1
## y_rep[223] 160.44 0.13 5.60 153.35 160.47 167.40 1996 1
## y_rep[224] 149.26 0.13 5.69 141.87 149.21 156.72 1976 1
## y_rep[225] 160.37 0.12 5.54 153.38 160.28 167.60 2015 1
## y_rep[226] 149.45 0.12 5.65 142.50 149.32 156.65 2092 1
## y_rep[227] 149.56 0.13 5.46 142.81 149.59 156.47 1691 1
## y_rep[228] 160.37 0.13 5.54 153.31 160.28 167.61 1916 1
## y_rep[229] 149.55 0.13 5.74 142.37 149.53 156.91 1994 1
## y_rep[230] 160.39 0.13 5.55 153.13 160.39 167.53 1770 1
## y_rep[231] 160.30 0.12 5.55 153.16 160.29 167.42 1978 1
## y_rep[232] 149.57 0.13 5.57 142.41 149.58 156.73 1833 1
## y_rep[233] 149.75 0.13 5.69 142.52 149.81 157.04 1848 1
## y_rep[234] 160.41 0.12 5.50 153.66 160.32 167.64 2041 1
## y_rep[235] 160.36 0.13 5.51 153.33 160.39 167.28 1903 1
## y_rep[236] 149.57 0.12 5.45 142.78 149.69 156.34 1956 1
## y_rep[237] 160.24 0.14 5.71 152.84 160.26 167.59 1742 1
## y_rep[238] 149.75 0.12 5.40 142.74 149.61 156.60 2105 1
## y_rep[239] 149.35 0.13 5.54 142.29 149.30 156.35 1837 1
## y_rep[240] 149.38 0.12 5.47 142.35 149.40 156.38 2158 1
## y_rep[241] 149.14 0.12 5.44 141.81 149.29 156.15 2185 1
## y_rep[242] 160.35 0.12 5.57 153.12 160.54 167.37 2019 1
## y_rep[243] 160.36 0.13 5.71 152.91 160.34 167.61 2041 1
## y_rep[244] 149.64 0.13 5.59 142.56 149.56 156.82 1992 1
## y_rep[245] 160.60 0.12 5.44 153.67 160.59 167.70 1978 1
## y_rep[246] 149.53 0.13 5.57 142.62 149.35 156.61 1944 1
## y_rep[247] 160.30 0.13 5.67 153.04 160.19 167.72 1978 1
## y_rep[248] 149.57 0.13 5.61 142.34 149.72 156.88 1962 1
## y_rep[249] 160.36 0.13 5.61 153.27 160.35 167.42 1961 1
## y_rep[250] 160.59 0.13 5.58 153.51 160.61 167.53 1913 1
## y_rep[251] 160.39 0.13 5.66 153.02 160.29 167.62 1880 1
## y_rep[252] 160.62 0.12 5.53 153.60 160.61 167.56 2041 1
## y_rep[253] 149.54 0.12 5.61 142.51 149.27 156.89 2051 1
## y_rep[254] 149.59 0.12 5.50 142.38 149.60 156.56 2069 1
## y_rep[255] 149.26 0.12 5.47 142.58 149.04 156.38 1980 1
## y_rep[256] 149.53 0.12 5.58 142.35 149.58 156.67 2155 1
## y_rep[257] 160.22 0.13 5.56 153.13 160.29 167.29 1871 1
## y_rep[258] 149.71 0.13 5.66 142.56 149.70 157.10 1999 1
## y_rep[259] 160.44 0.13 5.62 153.16 160.31 167.61 2001 1
## y_rep[260] 160.34 0.12 5.43 153.34 160.51 167.23 1992 1
## y_rep[261] 160.32 0.12 5.60 153.08 160.36 167.64 2135 1
## y_rep[262] 149.63 0.12 5.53 142.47 149.65 156.52 2039 1
## y_rep[263] 149.55 0.13 5.69 142.21 149.49 156.99 1997 1
## y_rep[264] 149.50 0.14 5.51 142.34 149.64 156.35 1619 1
## y_rep[265] 160.43 0.13 5.55 153.19 160.55 167.27 1895 1
## y_rep[266] 149.59 0.13 5.52 142.30 149.66 156.57 1946 1
## y_rep[267] 160.51 0.14 5.46 153.60 160.74 167.17 1519 1
## y_rep[268] 149.45 0.13 5.56 142.31 149.49 156.41 1843 1
## y_rep[269] 149.40 0.13 5.68 142.02 149.43 156.73 1947 1
## y_rep[270] 160.52 0.13 5.49 153.31 160.60 167.45 1910 1
## y_rep[271] 160.51 0.13 5.72 153.17 160.45 167.86 1951 1
## y_rep[272] 149.74 0.13 5.62 142.62 149.75 157.01 2015 1
## y_rep[273] 160.54 0.13 5.49 153.31 160.64 167.50 1855 1
## y_rep[274] 160.45 0.12 5.57 153.22 160.57 167.62 2010 1
## y_rep[275] 149.51 0.13 5.68 142.23 149.60 156.75 2031 1
## y_rep[276] 160.32 0.13 5.56 153.33 160.41 167.09 1886 1
## y_rep[277] 149.37 0.13 5.63 142.40 149.31 156.61 1987 1
## y_rep[278] 149.49 0.13 5.60 142.11 149.63 156.60 1993 1
## y_rep[279] 160.34 0.13 5.71 153.04 160.19 167.77 1943 1
## y_rep[280] 149.31 0.12 5.53 142.24 149.29 156.70 2034 1
## y_rep[281] 149.42 0.13 5.52 142.26 149.33 156.55 1845 1
## y_rep[282] 160.28 0.13 5.47 153.33 160.22 167.21 1819 1
## y_rep[283] 149.52 0.12 5.54 142.34 149.48 156.67 1992 1
## y_rep[284] 160.22 0.12 5.62 153.15 160.08 167.43 2053 1
## y_rep[285] 149.49 0.12 5.44 142.51 149.29 156.58 2131 1
## y_rep[286] 160.27 0.13 5.70 152.91 160.51 167.45 1984 1
## y_rep[287] 160.32 0.12 5.46 153.31 160.25 167.50 2045 1
## y_rep[288] 160.33 0.13 5.46 153.29 160.37 167.22 1664 1
## y_rep[289] 149.35 0.12 5.52 142.47 149.24 156.61 2090 1
## y_rep[290] 160.31 0.14 5.61 153.35 160.17 167.50 1685 1
## y_rep[291] 160.50 0.14 5.64 153.01 160.64 167.56 1746 1
## y_rep[292] 160.43 0.12 5.52 153.28 160.54 167.38 2053 1
## y_rep[293] 160.42 0.12 5.54 153.39 160.50 167.39 2059 1
## y_rep[294] 149.49 0.12 5.52 142.56 149.46 156.57 2048 1
## y_rep[295] 160.25 0.12 5.52 152.96 160.30 167.36 2022 1
## y_rep[296] 149.51 0.12 5.53 142.64 149.59 156.52 2145 1
## y_rep[297] 149.48 0.12 5.50 142.55 149.33 156.36 2113 1
## y_rep[298] 149.43 0.13 5.69 142.44 149.30 156.64 1927 1
## y_rep[299] 160.32 0.13 5.52 153.29 160.33 167.52 1854 1
## y_rep[300] 160.36 0.12 5.52 153.59 160.36 167.65 2081 1
## y_rep[301] 149.44 0.13 5.77 142.04 149.33 157.09 2056 1
## y_rep[302] 160.32 0.13 5.69 153.02 160.33 167.56 1937 1
## y_rep[303] 149.49 0.13 5.63 142.41 149.48 156.82 1954 1
## y_rep[304] 160.20 0.13 5.61 153.14 160.15 167.30 2000 1
## y_rep[305] 160.26 0.12 5.44 153.16 160.31 166.98 2085 1
## y_rep[306] 160.09 0.12 5.42 152.96 160.11 167.01 2020 1
## y_rep[307] 160.32 0.12 5.54 153.25 160.40 167.13 2161 1
## y_rep[308] 149.28 0.12 5.61 142.24 149.19 156.57 2027 1
## y_rep[309] 149.43 0.13 5.55 142.41 149.30 156.58 1800 1
## y_rep[310] 149.83 0.12 5.55 142.56 149.84 157.05 2078 1
## y_rep[311] 149.53 0.12 5.56 142.42 149.45 156.47 1988 1
## y_rep[312] 149.63 0.13 5.54 142.52 149.81 156.71 1855 1
## y_rep[313] 160.21 0.13 5.75 152.92 160.21 167.53 1867 1
## y_rep[314] 149.35 0.13 5.68 142.20 149.21 156.68 1999 1
## y_rep[315] 160.39 0.13 5.59 153.10 160.53 167.42 1855 1
## y_rep[316] 149.46 0.13 5.60 142.28 149.44 156.72 1955 1
## y_rep[317] 160.16 0.13 5.65 152.94 160.28 167.41 1921 1
## y_rep[318] 149.50 0.12 5.52 142.35 149.57 156.52 2083 1
## y_rep[319] 160.42 0.12 5.49 153.46 160.38 167.48 1971 1
## y_rep[320] 149.31 0.12 5.50 142.36 149.13 156.48 2005 1
## y_rep[321] 160.22 0.13 5.62 152.85 160.26 167.41 1857 1
## y_rep[322] 149.47 0.12 5.52 142.43 149.34 156.51 2091 1
## y_rep[323] 160.29 0.13 5.49 153.15 160.40 167.34 1840 1
## y_rep[324] 160.34 0.12 5.50 153.15 160.48 167.47 2053 1
## y_rep[325] 149.51 0.13 5.59 142.48 149.47 156.81 1948 1
## y_rep[326] 149.52 0.12 5.51 142.55 149.65 156.69 1995 1
## y_rep[327] 149.50 0.13 5.55 142.55 149.49 156.84 1884 1
## y_rep[328] 160.37 0.13 5.63 153.41 160.33 167.69 1961 1
## y_rep[329] 160.45 0.12 5.52 153.62 160.39 167.48 1978 1
## y_rep[330] 160.32 0.13 5.61 152.98 160.34 167.47 1943 1
## y_rep[331] 160.38 0.14 5.44 153.47 160.42 167.39 1473 1
## y_rep[332] 160.31 0.12 5.46 153.34 160.28 167.57 1931 1
## y_rep[333] 149.63 0.12 5.58 142.58 149.62 157.08 2080 1
## y_rep[334] 160.36 0.12 5.56 153.30 160.47 167.45 2068 1
## y_rep[335] 149.22 0.12 5.40 142.23 149.29 156.14 2059 1
## y_rep[336] 160.44 0.13 5.65 153.19 160.58 167.63 1973 1
## y_rep[337] 149.66 0.13 5.55 142.51 149.66 156.78 1949 1
## y_rep[338] 149.67 0.12 5.45 142.57 149.69 156.63 2008 1
## y_rep[339] 149.48 0.13 5.55 142.54 149.44 156.51 1920 1
## y_rep[340] 149.66 0.12 5.55 142.53 149.67 156.77 2011 1
## y_rep[341] 149.60 0.13 5.78 142.29 149.72 157.03 1998 1
## y_rep[342] 160.21 0.13 5.80 152.93 160.10 167.57 1945 1
## y_rep[343] 149.41 0.13 5.62 142.23 149.38 156.62 1806 1
## y_rep[344] 160.48 0.15 5.52 153.63 160.54 167.53 1420 1
## y_rep[345] 149.53 0.12 5.58 142.48 149.56 156.87 2025 1
## y_rep[346] 160.51 0.13 5.54 153.46 160.61 167.40 1795 1
## y_rep[347] 149.42 0.13 5.71 142.19 149.44 156.65 1918 1
## y_rep[348] 149.28 0.12 5.49 142.45 149.21 156.35 2088 1
## y_rep[349] 149.49 0.13 5.54 142.26 149.50 156.54 1859 1
## y_rep[350] 160.26 0.13 5.45 153.43 160.16 167.38 1808 1
## y_rep[351] 149.37 0.12 5.44 142.29 149.27 156.38 1939 1
## y_rep[352] 160.39 0.13 5.71 153.01 160.50 167.76 1953 1
## lp__ -779.74 0.04 1.24 -781.47 -779.44 -778.50 1045 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:55:46 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).Corresponds to a frequentist model with the intercept fixed to zero:
summary(lm(height~as.factor(male)-1,data=howell2))##
## Call:
## lm(formula = height ~ as.factor(male) - 1, data = howell2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20.6585 -3.4635 0.2965 3.4715 18.7115
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## as.factor(male)0 149.5135 0.4049 369.2 <2e-16 ***
## as.factor(male)1 160.3585 0.4311 372.0 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.537 on 350 degrees of freedom
## Multiple R-squared: 0.9987, Adjusted R-squared: 0.9987
## F-statistic: 1.374e+05 on 2 and 350 DF, p-value: < 2.2e-16Even if you use a index variable (with values 1 or 2), you can also use the same notation as for dummy variables
X <- model.matrix(~male.index, howell2)
m5.2.bis.data <- list(N = nrow(howell2), height = howell2$height, X = X, n = length(unique(howell2$male.index)))
str(m5.2.bis.data)## List of 4
## $ N : int 352
## $ height: num [1:352] 152 140 137 157 145 ...
## $ X : num [1:352, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:352] "1" "2" "3" "4" ...
## .. ..$ : chr [1:2] "(Intercept)" "male.index"
## ..- attr(*, "assign")= int [1:2] 0 1
## $ n : int 2data {
int<lower=0> N; // number of observations
int<lower=0> n; // number of levels of the categorical variable
vector[N] height; // response variables
matrix[N,n] X; // categorical variable
}
parameters {
vector[n] beta; // intercepts of the categorical variable
real sigma;
}
transformed parameters {
vector[N] mu;
mu = X*beta;
}
model {
height ~ normal(mu ,sigma); // likelihood
sigma ~ cauchy(0,5);
beta[1] ~ normal(178,20);
beta[2] ~ normal(0,10);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(mu[i], sigma);
}
fit.m5_2_bis <- sampling(m5.2.bis, data = m5.2.bis.data, iter = 1000, chains = 4, cores = 4)
print(fit.m5_2_bis, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: ea1df6c828dbad449e9a4c6c6321058c.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 138.80 0.03 0.90 137.61 138.82 139.97 724 1
## beta[2] 10.76 0.02 0.58 10.02 10.76 11.50 736 1
## sigma 5.57 0.01 0.21 5.30 5.55 5.84 942 1
## mu[1] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[2] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[3] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[4] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[5] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[6] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[7] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[8] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[9] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[10] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[11] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[12] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[13] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[14] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[15] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[16] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[17] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[18] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[19] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[20] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[21] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[22] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[23] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[24] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[25] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[26] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[27] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[28] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[29] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[30] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[31] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[32] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[33] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[34] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[35] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[36] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[37] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[38] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[39] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[40] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[41] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[42] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[43] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[44] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[45] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[46] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[47] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[48] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[49] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[50] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[51] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[52] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[53] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[54] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[55] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[56] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[57] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[58] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[59] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[60] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[61] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[62] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[63] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[64] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[65] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[66] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[67] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[68] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[69] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[70] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[71] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[72] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[73] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[74] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[75] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[76] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[77] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[78] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[79] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[80] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[81] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[82] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[83] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[84] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[85] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[86] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[87] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[88] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[89] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[90] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[91] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[92] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[93] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[94] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[95] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[96] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[97] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[98] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[99] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[100] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[101] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[102] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[103] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[104] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[105] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[106] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[107] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[108] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[109] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[110] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[111] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[112] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[113] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[114] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[115] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[116] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[117] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[118] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[119] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[120] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[121] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[122] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[123] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[124] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[125] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[126] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[127] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[128] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[129] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[130] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[131] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[132] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[133] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[134] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[135] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[136] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[137] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[138] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[139] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[140] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[141] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[142] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[143] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[144] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[145] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[146] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[147] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[148] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[149] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[150] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[151] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[152] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[153] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[154] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[155] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[156] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[157] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[158] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[159] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[160] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[161] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[162] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[163] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[164] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[165] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[166] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[167] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[168] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[169] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[170] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[171] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[172] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[173] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[174] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[175] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[176] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[177] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[178] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[179] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[180] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[181] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[182] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[183] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[184] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[185] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[186] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[187] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[188] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[189] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[190] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[191] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[192] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[193] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[194] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[195] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[196] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[197] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[198] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[199] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[200] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[201] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[202] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[203] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[204] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[205] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[206] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[207] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[208] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[209] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[210] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[211] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[212] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[213] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[214] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[215] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[216] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[217] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[218] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[219] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[220] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[221] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[222] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[223] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[224] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[225] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[226] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[227] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[228] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[229] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[230] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[231] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[232] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[233] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[234] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[235] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[236] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[237] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[238] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[239] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[240] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[241] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[242] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[243] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[244] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[245] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[246] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[247] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[248] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[249] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[250] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[251] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[252] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[253] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[254] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[255] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[256] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[257] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[258] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[259] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[260] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[261] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[262] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[263] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[264] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[265] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[266] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[267] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[268] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[269] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[270] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[271] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[272] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[273] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[274] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[275] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[276] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[277] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[278] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[279] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[280] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[281] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[282] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[283] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[284] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[285] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[286] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[287] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[288] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[289] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[290] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[291] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[292] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[293] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[294] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[295] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[296] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[297] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[298] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[299] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[300] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[301] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[302] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[303] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[304] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[305] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[306] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[307] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[308] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[309] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[310] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[311] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[312] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[313] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[314] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[315] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[316] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[317] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[318] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[319] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[320] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[321] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[322] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[323] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[324] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[325] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[326] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[327] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[328] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[329] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[330] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[331] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[332] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[333] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[334] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[335] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[336] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[337] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[338] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[339] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[340] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[341] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[342] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[343] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[344] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[345] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[346] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[347] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[348] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[349] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[350] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## mu[351] 149.56 0.01 0.40 149.02 149.56 150.08 956 1
## mu[352] 160.32 0.01 0.42 159.78 160.33 160.85 1327 1
## y_rep[1] 160.19 0.13 5.74 152.80 160.16 167.43 1963 1
## y_rep[2] 149.63 0.12 5.66 142.21 149.78 156.81 2058 1
## y_rep[3] 149.57 0.12 5.53 142.31 149.63 156.57 2020 1
## y_rep[4] 160.08 0.12 5.67 152.94 160.10 167.17 2069 1
## y_rep[5] 149.32 0.12 5.53 142.21 149.42 156.54 1963 1
## y_rep[6] 160.38 0.13 5.58 153.43 160.25 167.56 1863 1
## y_rep[7] 149.63 0.13 5.84 142.06 149.57 157.18 1956 1
## y_rep[8] 160.29 0.13 5.71 153.01 160.17 167.75 2012 1
## y_rep[9] 149.42 0.13 5.63 142.22 149.52 156.49 1866 1
## y_rep[10] 160.41 0.13 5.68 153.01 160.67 167.58 1967 1
## y_rep[11] 149.47 0.13 5.65 142.27 149.47 156.60 2018 1
## y_rep[12] 160.29 0.12 5.55 153.25 160.26 167.04 2036 1
## y_rep[13] 149.46 0.13 5.64 142.12 149.47 156.51 1952 1
## y_rep[14] 149.56 0.13 5.68 141.90 149.71 156.75 2025 1
## y_rep[15] 149.55 0.12 5.50 142.45 149.63 156.60 1964 1
## y_rep[16] 160.35 0.13 5.46 153.48 160.16 167.38 1894 1
## y_rep[17] 160.08 0.12 5.52 153.16 160.11 166.99 1959 1
## y_rep[18] 149.66 0.13 5.63 142.44 149.79 156.76 1904 1
## y_rep[19] 160.30 0.13 5.51 153.11 160.36 167.31 1931 1
## y_rep[20] 149.57 0.13 5.71 141.97 149.71 156.93 1885 1
## y_rep[21] 160.14 0.13 5.73 152.91 160.16 167.65 2022 1
## y_rep[22] 149.63 0.13 5.65 142.48 149.59 156.86 1944 1
## y_rep[23] 149.50 0.12 5.59 142.21 149.56 156.80 2063 1
## y_rep[24] 160.34 0.13 5.50 153.20 160.41 167.24 1891 1
## y_rep[25] 160.43 0.13 5.61 153.18 160.48 167.54 2009 1
## y_rep[26] 149.51 0.12 5.65 142.32 149.45 156.72 2061 1
## y_rep[27] 149.61 0.13 5.56 142.37 149.65 156.80 1916 1
## y_rep[28] 149.66 0.12 5.63 142.52 149.61 156.79 2049 1
## y_rep[29] 149.42 0.13 5.51 142.54 149.29 156.66 1904 1
## y_rep[30] 149.78 0.13 5.49 142.82 149.82 156.73 1805 1
## y_rep[31] 149.43 0.12 5.58 142.12 149.60 156.37 1998 1
## y_rep[32] 160.20 0.13 5.67 153.12 160.14 167.45 1879 1
## y_rep[33] 160.25 0.13 5.65 153.01 160.15 167.69 1979 1
## y_rep[34] 160.42 0.12 5.64 153.23 160.29 167.87 2040 1
## y_rep[35] 160.17 0.13 5.56 153.14 160.21 167.28 1731 1
## y_rep[36] 149.75 0.13 5.59 142.63 149.60 157.05 1991 1
## y_rep[37] 149.59 0.13 5.66 142.38 149.69 156.80 1889 1
## y_rep[38] 160.39 0.13 5.50 153.56 160.24 167.35 1890 1
## y_rep[39] 149.61 0.12 5.49 142.53 149.67 156.60 1964 1
## y_rep[40] 160.39 0.12 5.57 153.51 160.11 167.62 2130 1
## y_rep[41] 160.27 0.12 5.53 153.12 160.34 167.09 1983 1
## y_rep[42] 160.35 0.12 5.50 153.61 160.19 167.23 2101 1
## y_rep[43] 149.56 0.12 5.46 142.61 149.48 156.42 1978 1
## y_rep[44] 149.75 0.13 5.55 142.79 149.82 156.77 1902 1
## y_rep[45] 149.90 0.13 5.56 142.64 149.94 157.14 1710 1
## y_rep[46] 149.58 0.12 5.52 142.67 149.60 156.65 2015 1
## y_rep[47] 160.27 0.13 5.50 153.25 160.33 167.33 1733 1
## y_rep[48] 149.57 0.12 5.56 142.49 149.48 156.78 2062 1
## y_rep[49] 160.37 0.12 5.58 153.28 160.48 167.31 2047 1
## y_rep[50] 149.60 0.12 5.54 142.41 149.67 156.61 2026 1
## y_rep[51] 149.70 0.12 5.55 142.63 149.60 156.77 2069 1
## y_rep[52] 149.62 0.13 5.55 142.36 149.77 156.55 1943 1
## y_rep[53] 160.15 0.13 5.56 153.20 160.17 167.15 1932 1
## y_rep[54] 160.41 0.12 5.57 153.18 160.44 167.49 2000 1
## y_rep[55] 149.66 0.14 5.49 142.77 149.75 156.87 1467 1
## y_rep[56] 160.15 0.13 5.61 153.06 160.00 167.46 1802 1
## y_rep[57] 149.79 0.12 5.63 142.67 149.64 156.97 2203 1
## y_rep[58] 160.31 0.12 5.44 153.43 160.33 167.39 1977 1
## y_rep[59] 149.87 0.13 5.69 142.50 149.94 157.08 1920 1
## y_rep[60] 160.25 0.13 5.63 152.93 160.31 167.58 1983 1
## y_rep[61] 149.46 0.13 5.42 142.64 149.52 156.28 1759 1
## y_rep[62] 149.55 0.12 5.56 142.51 149.59 156.60 2016 1
## y_rep[63] 160.23 0.12 5.57 152.83 160.31 167.30 2094 1
## y_rep[64] 160.13 0.13 5.72 152.65 160.19 167.33 1884 1
## y_rep[65] 149.53 0.12 5.50 142.49 149.62 156.50 2186 1
## y_rep[66] 160.30 0.13 5.70 153.27 160.24 167.69 2028 1
## y_rep[67] 149.50 0.12 5.67 142.06 149.64 156.47 2114 1
## y_rep[68] 149.77 0.13 5.56 142.85 149.95 156.61 1942 1
## y_rep[69] 160.55 0.13 5.46 153.70 160.63 167.37 1860 1
## y_rep[70] 149.57 0.13 5.64 142.50 149.55 156.92 1931 1
## y_rep[71] 160.39 0.13 5.58 153.27 160.36 167.60 1908 1
## y_rep[72] 160.40 0.12 5.59 153.15 160.51 167.52 2005 1
## y_rep[73] 160.45 0.12 5.68 153.07 160.52 167.70 2181 1
## y_rep[74] 149.33 0.12 5.67 142.16 149.55 156.60 2094 1
## y_rep[75] 149.46 0.12 5.59 142.40 149.50 156.69 2044 1
## y_rep[76] 149.21 0.12 5.53 142.09 149.32 156.28 2036 1
## y_rep[77] 149.40 0.14 5.65 142.07 149.54 156.76 1686 1
## y_rep[78] 160.33 0.13 5.58 153.05 160.36 167.35 1778 1
## y_rep[79] 149.35 0.13 5.73 142.07 149.31 156.76 1822 1
## y_rep[80] 149.64 0.13 5.67 142.69 149.51 156.87 1948 1
## y_rep[81] 160.47 0.12 5.60 153.09 160.54 167.71 2067 1
## y_rep[82] 160.33 0.13 5.63 153.33 160.29 167.52 1759 1
## y_rep[83] 149.61 0.12 5.51 142.67 149.67 156.56 1965 1
## y_rep[84] 160.30 0.12 5.57 153.18 160.24 167.81 2112 1
## y_rep[85] 149.53 0.13 5.68 142.40 149.47 156.94 1910 1
## y_rep[86] 160.46 0.14 5.69 153.34 160.36 167.80 1745 1
## y_rep[87] 149.76 0.13 5.64 142.71 149.93 156.88 1888 1
## y_rep[88] 160.42 0.13 5.68 153.00 160.34 167.81 1950 1
## y_rep[89] 149.53 0.13 5.60 142.13 149.49 156.60 1925 1
## y_rep[90] 160.26 0.13 5.63 152.92 160.29 167.60 1965 1
## y_rep[91] 160.30 0.13 5.60 153.01 160.48 167.35 1849 1
## y_rep[92] 160.38 0.13 5.51 153.51 160.32 167.45 1942 1
## y_rep[93] 149.66 0.13 5.62 142.55 149.55 156.97 1986 1
## y_rep[94] 149.52 0.13 5.57 142.37 149.38 156.92 1861 1
## y_rep[95] 160.24 0.12 5.63 153.03 160.31 167.36 2055 1
## y_rep[96] 160.21 0.12 5.50 153.00 160.09 167.29 1960 1
## y_rep[97] 149.40 0.12 5.71 142.05 149.73 156.63 2113 1
## y_rep[98] 160.58 0.12 5.67 153.32 160.46 167.84 2058 1
## y_rep[99] 160.24 0.14 5.42 153.07 160.35 167.05 1586 1
## y_rep[100] 149.35 0.12 5.39 142.52 149.11 156.24 1878 1
## y_rep[101] 160.24 0.14 5.68 152.78 160.34 167.35 1670 1
## y_rep[102] 149.52 0.13 5.66 142.08 149.56 156.73 1886 1
## y_rep[103] 149.58 0.13 5.68 142.27 149.46 156.85 1815 1
## y_rep[104] 160.31 0.12 5.54 153.20 160.33 167.33 1993 1
## y_rep[105] 160.29 0.12 5.64 153.20 160.24 167.23 2097 1
## y_rep[106] 149.53 0.13 5.66 142.32 149.64 156.40 1916 1
## y_rep[107] 149.45 0.12 5.71 142.10 149.34 156.80 2093 1
## y_rep[108] 149.56 0.12 5.58 142.41 149.30 156.79 2020 1
## y_rep[109] 160.42 0.13 5.61 153.34 160.67 167.24 1973 1
## y_rep[110] 149.75 0.12 5.48 142.82 149.76 156.71 1947 1
## y_rep[111] 160.45 0.12 5.57 153.24 160.26 167.61 2044 1
## y_rep[112] 149.76 0.12 5.45 142.79 149.80 156.87 2045 1
## y_rep[113] 149.56 0.13 5.52 142.44 149.52 156.68 1851 1
## y_rep[114] 160.04 0.13 5.69 152.70 160.06 167.27 1992 1
## y_rep[115] 160.28 0.13 5.60 152.88 160.23 167.54 1783 1
## y_rep[116] 149.72 0.13 5.54 142.53 149.84 156.82 1891 1
## y_rep[117] 160.51 0.12 5.54 153.61 160.57 167.65 1976 1
## y_rep[118] 160.39 0.12 5.47 153.39 160.47 167.42 1979 1
## y_rep[119] 149.62 0.14 5.65 142.62 149.38 156.91 1751 1
## y_rep[120] 149.41 0.13 5.63 142.26 149.38 156.60 1954 1
## y_rep[121] 160.45 0.12 5.62 153.15 160.54 167.78 2079 1
## y_rep[122] 149.46 0.13 5.62 142.31 149.41 156.71 1914 1
## y_rep[123] 160.22 0.13 5.68 152.88 160.18 167.37 2022 1
## y_rep[124] 149.70 0.13 5.69 142.55 149.68 157.11 1904 1
## y_rep[125] 149.59 0.13 5.65 142.56 149.51 157.17 1957 1
## y_rep[126] 149.48 0.13 5.43 142.32 149.61 156.34 1731 1
## y_rep[127] 149.53 0.13 5.66 142.14 149.52 156.90 1961 1
## y_rep[128] 160.32 0.12 5.51 153.07 160.23 167.31 1950 1
## y_rep[129] 149.55 0.13 5.45 142.35 149.55 156.49 1731 1
## y_rep[130] 149.50 0.12 5.69 142.28 149.37 156.88 2246 1
## y_rep[131] 160.21 0.12 5.50 153.09 160.29 167.42 1939 1
## y_rep[132] 160.44 0.14 5.62 153.23 160.51 167.74 1702 1
## y_rep[133] 149.63 0.13 5.67 142.39 149.65 156.66 1810 1
## y_rep[134] 160.37 0.13 5.56 153.29 160.53 167.44 1916 1
## y_rep[135] 149.42 0.13 5.58 142.33 149.36 156.62 1857 1
## y_rep[136] 160.24 0.13 5.56 152.93 160.34 167.35 1960 1
## y_rep[137] 160.31 0.12 5.59 153.26 160.33 167.53 2100 1
## y_rep[138] 160.33 0.13 5.71 152.75 160.26 167.73 1978 1
## y_rep[139] 149.79 0.13 5.58 142.76 149.71 157.07 1931 1
## y_rep[140] 160.18 0.13 5.53 153.16 160.05 167.31 1904 1
## y_rep[141] 149.41 0.13 5.61 142.15 149.36 156.59 1931 1
## y_rep[142] 149.77 0.13 5.73 142.22 149.93 156.93 1817 1
## y_rep[143] 149.73 0.12 5.52 142.54 149.69 156.81 2126 1
## y_rep[144] 149.56 0.13 5.56 142.60 149.51 156.71 1946 1
## y_rep[145] 149.38 0.12 5.55 142.30 149.39 156.57 2086 1
## y_rep[146] 160.21 0.12 5.38 153.25 160.28 167.07 1980 1
## y_rep[147] 149.34 0.13 5.86 141.64 149.40 156.84 1905 1
## y_rep[148] 149.55 0.13 5.61 142.45 149.55 156.90 1919 1
## y_rep[149] 160.19 0.13 5.58 153.02 160.15 167.35 1965 1
## y_rep[150] 149.70 0.12 5.60 142.68 149.61 156.93 2127 1
## y_rep[151] 160.41 0.13 5.51 153.23 160.42 167.40 1939 1
## y_rep[152] 149.67 0.13 5.49 142.74 149.59 156.42 1901 1
## y_rep[153] 160.34 0.13 5.53 153.29 160.32 167.64 1860 1
## y_rep[154] 149.72 0.12 5.58 142.49 149.95 156.94 2165 1
## y_rep[155] 160.29 0.12 5.52 153.15 160.45 167.37 2199 1
## y_rep[156] 149.51 0.12 5.51 142.47 149.55 156.55 2039 1
## y_rep[157] 149.63 0.12 5.49 142.50 149.61 156.75 2135 1
## y_rep[158] 160.48 0.13 5.64 153.34 160.40 167.75 1902 1
## y_rep[159] 160.53 0.13 5.69 153.33 160.55 167.59 2023 1
## y_rep[160] 149.34 0.13 5.57 142.29 149.10 156.56 1927 1
## y_rep[161] 160.26 0.13 5.69 152.90 160.28 167.62 1987 1
## y_rep[162] 149.42 0.13 5.60 142.22 149.47 156.65 1897 1
## y_rep[163] 160.30 0.13 5.60 153.22 160.33 167.49 1891 1
## y_rep[164] 160.14 0.12 5.59 153.00 160.00 167.31 2068 1
## y_rep[165] 160.22 0.12 5.34 153.40 160.28 166.92 2060 1
## y_rep[166] 149.55 0.13 5.53 142.59 149.47 156.76 1742 1
## y_rep[167] 160.43 0.13 5.64 152.85 160.59 167.60 1847 1
## y_rep[168] 149.51 0.12 5.57 142.19 149.52 156.58 2040 1
## y_rep[169] 149.62 0.13 5.47 142.75 149.55 156.66 1880 1
## y_rep[170] 149.57 0.13 5.58 142.30 149.68 156.60 1954 1
## y_rep[171] 149.79 0.12 5.51 142.47 149.90 156.82 2115 1
## y_rep[172] 160.32 0.12 5.61 153.09 160.40 167.34 2026 1
## y_rep[173] 149.59 0.12 5.57 142.26 149.56 156.77 2041 1
## y_rep[174] 160.18 0.12 5.61 152.92 160.22 167.30 2039 1
## y_rep[175] 149.64 0.12 5.73 142.19 149.68 156.86 2114 1
## y_rep[176] 160.56 0.13 5.66 153.17 160.54 168.05 1923 1
## y_rep[177] 149.62 0.13 5.59 142.57 149.61 156.76 1904 1
## y_rep[178] 149.71 0.12 5.53 142.57 149.71 156.95 2017 1
## y_rep[179] 160.40 0.13 5.61 153.23 160.41 167.54 2001 1
## y_rep[180] 149.72 0.13 5.63 142.41 149.85 156.75 1902 1
## y_rep[181] 160.27 0.12 5.60 153.22 160.28 167.52 2013 1
## y_rep[182] 149.56 0.12 5.58 142.15 149.57 156.63 2228 1
## y_rep[183] 149.50 0.13 5.56 142.32 149.41 156.56 1733 1
## y_rep[184] 149.65 0.12 5.51 142.81 149.46 156.87 1980 1
## y_rep[185] 160.40 0.13 5.63 153.21 160.39 167.59 1961 1
## y_rep[186] 160.13 0.12 5.59 153.03 160.18 167.28 2082 1
## y_rep[187] 149.43 0.13 5.65 142.20 149.60 156.77 1981 1
## y_rep[188] 160.27 0.12 5.58 153.16 160.29 167.38 2036 1
## y_rep[189] 149.44 0.13 5.49 142.39 149.41 156.67 1829 1
## y_rep[190] 160.50 0.13 5.60 153.17 160.47 167.58 1958 1
## y_rep[191] 149.58 0.12 5.46 142.67 149.54 156.67 1965 1
## y_rep[192] 160.38 0.12 5.51 153.18 160.45 167.22 1992 1
## y_rep[193] 149.63 0.13 5.55 142.53 149.69 156.75 1907 1
## y_rep[194] 160.29 0.13 5.61 153.14 160.30 167.54 1880 1
## y_rep[195] 149.50 0.13 5.60 142.50 149.44 157.13 1903 1
## y_rep[196] 149.49 0.13 5.64 142.16 149.57 156.80 1945 1
## y_rep[197] 149.33 0.12 5.40 142.46 149.13 156.24 2019 1
## y_rep[198] 160.52 0.13 5.60 153.39 160.49 167.69 1962 1
## y_rep[199] 160.19 0.13 5.78 152.94 160.12 167.57 1847 1
## y_rep[200] 149.62 0.13 5.62 142.26 149.71 156.76 1997 1
## y_rep[201] 160.26 0.12 5.61 153.14 160.18 167.43 2088 1
## y_rep[202] 149.66 0.13 5.49 142.53 149.61 156.92 1844 1
## y_rep[203] 160.33 0.13 5.62 153.28 160.28 167.64 1768 1
## y_rep[204] 149.60 0.13 5.67 142.29 149.63 156.87 1948 1
## y_rep[205] 160.41 0.12 5.45 153.45 160.46 167.40 2034 1
## y_rep[206] 149.56 0.13 5.57 142.43 149.58 156.75 1921 1
## y_rep[207] 149.50 0.12 5.56 142.70 149.53 156.51 2213 1
## y_rep[208] 149.77 0.12 5.52 142.78 149.72 156.78 1959 1
## y_rep[209] 149.50 0.13 5.63 142.42 149.53 156.73 1879 1
## y_rep[210] 149.61 0.12 5.50 142.44 149.61 156.57 2007 1
## y_rep[211] 160.24 0.13 5.59 152.96 160.35 167.28 1977 1
## y_rep[212] 160.27 0.13 5.74 153.01 160.15 167.76 2018 1
## y_rep[213] 160.27 0.12 5.43 153.34 160.26 167.00 1926 1
## y_rep[214] 149.54 0.12 5.45 142.64 149.60 156.49 2018 1
## y_rep[215] 160.45 0.13 5.60 153.41 160.44 167.58 1913 1
## y_rep[216] 160.35 0.13 5.75 152.82 160.25 167.68 2012 1
## y_rep[217] 160.10 0.12 5.56 152.80 160.06 167.06 1981 1
## y_rep[218] 149.66 0.13 5.65 142.49 149.78 156.77 1961 1
## y_rep[219] 160.25 0.13 5.57 153.33 160.13 167.23 1828 1
## y_rep[220] 149.49 0.13 5.55 142.33 149.42 156.80 1703 1
## y_rep[221] 149.50 0.12 5.58 142.35 149.55 156.56 2247 1
## y_rep[222] 149.63 0.13 5.66 142.31 149.49 156.89 1986 1
## y_rep[223] 160.36 0.13 5.51 153.07 160.23 167.62 1915 1
## y_rep[224] 149.33 0.13 5.56 142.08 149.27 156.50 1901 1
## y_rep[225] 160.39 0.14 5.63 153.24 160.30 167.38 1662 1
## y_rep[226] 149.58 0.12 5.50 142.44 149.79 156.48 2083 1
## y_rep[227] 149.52 0.13 5.67 142.22 149.42 156.77 2002 1
## y_rep[228] 160.26 0.13 5.68 153.23 160.34 167.52 1897 1
## y_rep[229] 149.60 0.13 5.59 142.27 149.62 156.86 1882 1
## y_rep[230] 160.20 0.12 5.59 152.97 160.13 167.33 2059 1
## y_rep[231] 160.26 0.12 5.44 153.15 160.29 167.13 1954 1
## y_rep[232] 149.61 0.12 5.52 142.57 149.71 156.50 2117 1
## y_rep[233] 149.50 0.12 5.46 142.50 149.47 156.60 2069 1
## y_rep[234] 160.29 0.12 5.66 152.90 160.37 167.51 2152 1
## y_rep[235] 160.27 0.13 5.63 153.28 160.31 167.36 1812 1
## y_rep[236] 149.72 0.14 5.63 142.75 149.89 156.98 1690 1
## y_rep[237] 160.19 0.12 5.57 152.85 160.09 167.47 2257 1
## y_rep[238] 149.37 0.12 5.55 142.20 149.50 156.28 2088 1
## y_rep[239] 149.49 0.12 5.52 142.53 149.51 156.55 2037 1
## y_rep[240] 149.66 0.13 5.59 142.33 149.60 156.72 1878 1
## y_rep[241] 149.73 0.12 5.56 142.60 149.62 156.86 2108 1
## y_rep[242] 160.40 0.13 5.73 152.99 160.30 167.81 2012 1
## y_rep[243] 160.27 0.13 5.43 153.24 160.24 167.29 1795 1
## y_rep[244] 149.61 0.13 5.60 142.65 149.38 156.91 1976 1
## y_rep[245] 160.28 0.12 5.54 153.25 160.10 167.37 2114 1
## y_rep[246] 149.55 0.14 5.68 142.05 149.47 156.81 1753 1
## y_rep[247] 160.18 0.13 5.55 153.19 160.22 167.42 1797 1
## y_rep[248] 149.48 0.13 5.62 142.46 149.58 156.70 1923 1
## y_rep[249] 160.41 0.13 5.69 153.05 160.33 167.59 2071 1
## y_rep[250] 160.22 0.13 5.57 153.28 160.26 167.26 1926 1
## y_rep[251] 160.50 0.13 5.53 153.46 160.41 167.63 1753 1
## y_rep[252] 160.01 0.13 5.67 152.96 159.95 167.28 1971 1
## y_rep[253] 149.52 0.13 5.55 142.38 149.55 156.70 1843 1
## y_rep[254] 149.66 0.13 5.57 142.43 149.58 156.92 1931 1
## y_rep[255] 149.45 0.13 5.63 142.16 149.46 156.53 1850 1
## y_rep[256] 149.66 0.13 5.48 142.78 149.68 156.52 1877 1
## y_rep[257] 160.20 0.12 5.49 153.38 160.13 167.30 2031 1
## y_rep[258] 149.47 0.13 5.63 142.23 149.54 156.45 2025 1
## y_rep[259] 160.24 0.12 5.48 153.26 160.31 167.20 2122 1
## y_rep[260] 160.33 0.13 5.70 153.20 160.07 167.57 2027 1
## y_rep[261] 160.31 0.13 5.56 153.13 160.34 167.27 1930 1
## y_rep[262] 149.58 0.12 5.43 142.50 149.53 156.35 1920 1
## y_rep[263] 149.54 0.13 5.69 142.25 149.41 156.63 1963 1
## y_rep[264] 149.35 0.12 5.54 141.99 149.50 156.46 2177 1
## y_rep[265] 160.16 0.13 5.54 152.89 160.29 167.24 1866 1
## y_rep[266] 149.70 0.13 5.54 142.67 149.65 156.85 1810 1
## y_rep[267] 160.05 0.12 5.43 153.19 160.04 167.22 1914 1
## y_rep[268] 149.56 0.13 5.53 142.51 149.60 156.53 1762 1
## y_rep[269] 149.45 0.13 5.60 142.33 149.45 156.47 1917 1
## y_rep[270] 160.05 0.13 5.56 152.94 160.06 167.10 1722 1
## y_rep[271] 160.42 0.13 5.61 153.20 160.38 167.62 1800 1
## y_rep[272] 149.41 0.13 5.75 142.05 149.53 156.63 2059 1
## y_rep[273] 160.25 0.13 5.61 152.99 160.28 167.27 2013 1
## y_rep[274] 160.14 0.14 5.37 153.32 160.30 166.85 1563 1
## y_rep[275] 149.46 0.14 5.38 142.74 149.42 156.34 1545 1
## y_rep[276] 160.16 0.13 5.62 152.87 160.23 167.26 1839 1
## y_rep[277] 149.68 0.12 5.61 142.37 149.70 156.92 2094 1
## y_rep[278] 149.63 0.12 5.54 142.50 149.85 156.61 2169 1
## y_rep[279] 160.32 0.12 5.67 153.02 160.36 167.48 2134 1
## y_rep[280] 149.67 0.13 5.58 142.43 149.67 156.70 1906 1
## y_rep[281] 149.47 0.13 5.48 142.46 149.25 156.72 1904 1
## y_rep[282] 160.62 0.13 5.69 153.38 160.46 168.03 2067 1
## y_rep[283] 149.39 0.13 5.64 142.09 149.38 156.62 2005 1
## y_rep[284] 160.37 0.13 5.63 153.32 160.23 167.62 1833 1
## y_rep[285] 149.50 0.12 5.45 142.60 149.46 156.49 2071 1
## y_rep[286] 160.31 0.12 5.65 152.96 160.39 167.60 2062 1
## y_rep[287] 160.33 0.13 5.53 153.25 160.33 167.42 1851 1
## y_rep[288] 160.30 0.12 5.50 153.25 160.35 167.10 2166 1
## y_rep[289] 149.47 0.13 5.62 142.10 149.55 156.54 1910 1
## y_rep[290] 160.23 0.13 5.48 153.21 160.14 167.46 1747 1
## y_rep[291] 160.28 0.13 5.61 153.12 160.32 167.44 1959 1
## y_rep[292] 160.30 0.13 5.54 153.44 160.16 167.68 1961 1
## y_rep[293] 160.45 0.13 5.65 153.41 160.13 167.89 2025 1
## y_rep[294] 149.35 0.13 5.54 142.14 149.35 156.20 1958 1
## y_rep[295] 160.27 0.12 5.42 153.40 160.20 167.12 1979 1
## y_rep[296] 149.51 0.12 5.58 142.23 149.56 156.82 2030 1
## y_rep[297] 149.68 0.13 5.54 142.74 149.48 156.90 1958 1
## y_rep[298] 149.51 0.13 5.51 142.42 149.49 156.48 1931 1
## y_rep[299] 160.11 0.13 5.44 153.19 159.93 167.37 1786 1
## y_rep[300] 160.34 0.12 5.55 153.13 160.27 167.36 2065 1
## y_rep[301] 149.55 0.13 5.59 142.42 149.46 156.73 1822 1
## y_rep[302] 160.24 0.12 5.54 152.94 160.30 167.29 2160 1
## y_rep[303] 149.69 0.13 5.51 142.50 149.60 156.75 1713 1
## y_rep[304] 160.41 0.12 5.40 153.51 160.61 167.07 1965 1
## y_rep[305] 160.35 0.13 5.62 153.21 160.28 167.64 1912 1
## y_rep[306] 160.13 0.12 5.43 153.17 160.19 167.29 1946 1
## y_rep[307] 160.29 0.12 5.44 153.13 160.38 167.25 1965 1
## y_rep[308] 149.70 0.12 5.50 142.66 149.68 156.79 1993 1
## y_rep[309] 149.70 0.12 5.57 142.37 149.67 156.95 2053 1
## y_rep[310] 149.50 0.13 5.56 142.63 149.41 156.74 1836 1
## y_rep[311] 149.58 0.13 5.66 142.23 149.59 156.80 1803 1
## y_rep[312] 149.42 0.13 5.49 142.42 149.46 156.56 1854 1
## y_rep[313] 160.51 0.13 5.80 153.06 160.52 167.91 2051 1
## y_rep[314] 149.38 0.13 5.64 142.24 149.27 156.52 1929 1
## y_rep[315] 160.20 0.13 5.47 153.15 160.29 167.25 1736 1
## y_rep[316] 149.52 0.12 5.43 142.62 149.48 156.51 1994 1
## y_rep[317] 160.16 0.13 5.45 153.12 160.21 166.99 1858 1
## y_rep[318] 149.47 0.13 5.70 142.36 149.44 156.95 1925 1
## y_rep[319] 160.47 0.13 5.64 152.99 160.54 167.54 2015 1
## y_rep[320] 149.52 0.13 5.51 142.73 149.61 156.61 1919 1
## y_rep[321] 160.36 0.13 5.69 153.16 160.27 167.85 1971 1
## y_rep[322] 149.60 0.12 5.39 142.85 149.63 156.46 1977 1
## y_rep[323] 160.47 0.12 5.50 153.26 160.47 167.61 2197 1
## y_rep[324] 160.52 0.13 5.56 153.26 160.52 167.61 1939 1
## y_rep[325] 149.41 0.14 5.53 142.05 149.23 156.34 1655 1
## y_rep[326] 149.54 0.13 5.69 142.55 149.51 156.90 1889 1
## y_rep[327] 149.74 0.12 5.53 142.70 149.88 156.61 1969 1
## y_rep[328] 160.40 0.13 5.50 153.24 160.43 167.25 1859 1
## y_rep[329] 160.30 0.13 5.70 153.07 160.39 167.59 2049 1
## y_rep[330] 160.29 0.13 5.55 153.27 160.27 167.47 1847 1
## y_rep[331] 160.32 0.13 5.71 153.06 160.42 167.47 1971 1
## y_rep[332] 160.24 0.12 5.53 153.06 160.27 167.38 2120 1
## y_rep[333] 149.46 0.13 5.56 142.29 149.44 156.43 1903 1
## y_rep[334] 160.24 0.13 5.55 153.15 160.08 167.40 1884 1
## y_rep[335] 149.53 0.12 5.53 142.40 149.59 156.58 2095 1
## y_rep[336] 160.42 0.13 5.75 152.92 160.42 167.61 1865 1
## y_rep[337] 149.52 0.12 5.47 142.65 149.47 156.63 2130 1
## y_rep[338] 149.59 0.13 5.61 142.31 149.67 156.89 2017 1
## y_rep[339] 149.72 0.12 5.50 142.73 149.65 156.72 1985 1
## y_rep[340] 149.37 0.12 5.49 142.30 149.50 156.22 1932 1
## y_rep[341] 149.70 0.12 5.50 142.79 149.75 156.80 1946 1
## y_rep[342] 160.14 0.12 5.49 153.08 160.18 167.26 2038 1
## y_rep[343] 149.36 0.13 5.59 142.30 149.30 156.45 1909 1
## y_rep[344] 160.44 0.13 5.48 153.40 160.57 167.25 1847 1
## y_rep[345] 149.65 0.13 5.61 142.64 149.68 156.73 1826 1
## y_rep[346] 160.38 0.12 5.58 153.37 160.32 167.49 1996 1
## y_rep[347] 149.71 0.14 5.68 142.59 149.56 157.32 1692 1
## y_rep[348] 149.42 0.12 5.55 142.39 149.45 156.49 1994 1
## y_rep[349] 149.51 0.12 5.51 142.52 149.53 156.48 2011 1
## y_rep[350] 160.22 0.13 5.75 152.74 160.25 167.46 2093 1
## y_rep[351] 149.74 0.14 5.60 142.46 149.84 156.91 1513 1
## y_rep[352] 160.17 0.13 5.69 152.88 160.17 167.61 1844 1
## lp__ -782.21 0.04 1.17 -783.79 -781.92 -781.02 866 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:56:38 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).5 Polynomial regression
5.1 Howell 2: subset with only individuals over 18 years old
Standardizing the explanatory variable:
howell2 <- howell2 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))Data in a list:
m6.data <- list(N = nrow(howell2), height = howell2$height, weight = howell2$weight.z)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[3]; // parameters: intercept and slope
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ normal(0, 100); // prior of the slope
beta[3] ~ normal(0, 100);
sigma ~ uniform(0,50);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}fit.m6 <- sampling(m6, data = m6.data, iter = 1000, chains = 4, cores = 4)
print(fit.m6, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 154.61 0.01 0.36 154.15 154.61 155.07 1018 1.00
## beta[2] 5.85 0.01 0.27 5.51 5.86 6.20 799 1.00
## beta[3] -0.01 0.01 0.22 -0.29 -0.02 0.29 988 1.00
## sigma 5.11 0.00 0.19 4.86 5.11 5.35 2249 1.00
## y_rep[1] 157.25 0.11 5.19 150.72 157.24 163.86 2041 1.00
## y_rep[2] 146.94 0.11 5.07 140.51 146.86 153.45 1995 1.00
## y_rep[3] 142.78 0.12 5.20 135.99 142.81 149.52 1889 1.00
## y_rep[4] 161.85 0.12 5.14 155.16 161.72 168.38 1922 1.00
## y_rep[5] 151.37 0.11 5.10 144.74 151.47 157.58 2054 1.00
## y_rep[6] 170.67 0.13 5.44 163.83 170.58 177.49 1839 1.00
## y_rep[7] 148.26 0.12 5.15 141.63 148.17 154.92 1964 1.00
## y_rep[8] 164.10 0.12 5.36 157.28 164.15 170.81 2072 1.00
## y_rep[9] 145.60 0.14 5.27 138.80 145.65 152.30 1488 1.00
## y_rep[10] 163.15 0.12 5.16 156.48 163.23 169.62 1864 1.00
## y_rep[11] 159.00 0.11 5.14 152.53 158.95 165.47 2010 1.00
## y_rep[12] 151.00 0.12 4.93 144.76 150.94 157.60 1793 1.00
## y_rep[13] 146.66 0.12 5.11 140.38 146.39 153.40 1794 1.00
## y_rep[14] 157.26 0.12 5.23 150.44 157.38 164.06 1880 1.00
## y_rep[15] 144.38 0.12 5.19 137.47 144.58 150.91 1735 1.00
## y_rep[16] 157.90 0.11 5.13 151.33 157.97 164.51 2031 1.00
## y_rep[17] 152.10 0.12 5.23 145.43 152.05 158.93 2056 1.00
## y_rep[18] 148.43 0.12 5.19 141.73 148.61 155.05 1819 1.00
## y_rep[19] 158.27 0.12 5.18 151.63 158.26 164.72 1868 1.00
## y_rep[20] 152.75 0.13 5.24 146.04 152.81 159.40 1705 1.00
## y_rep[21] 145.87 0.11 4.98 139.40 145.92 152.12 1972 1.00
## y_rep[22] 148.24 0.11 5.14 141.53 148.24 154.90 2086 1.00
## y_rep[23] 146.09 0.12 5.11 139.69 146.11 152.79 1871 1.00
## y_rep[24] 150.29 0.12 5.26 143.52 150.16 157.00 1927 1.00
## y_rep[25] 163.76 0.11 5.09 157.31 163.75 170.35 2004 1.00
## y_rep[26] 147.85 0.11 5.13 141.35 147.74 154.31 1994 1.00
## y_rep[27] 148.78 0.12 5.21 142.38 148.76 155.28 2043 1.00
## y_rep[28] 156.10 0.11 4.95 149.77 156.12 162.42 2005 1.00
## y_rep[29] 148.80 0.12 5.20 141.98 148.84 155.20 1996 1.00
## y_rep[30] 146.01 0.12 5.17 139.14 146.14 152.38 1861 1.00
## y_rep[31] 156.62 0.13 5.27 149.85 156.66 163.40 1540 1.01
## y_rep[32] 157.04 0.11 5.09 150.65 156.91 163.45 2011 1.00
## y_rep[33] 158.49 0.12 5.19 151.85 158.50 165.07 1925 1.00
## y_rep[34] 158.64 0.11 5.07 152.08 158.72 165.00 2104 1.00
## y_rep[35] 165.13 0.12 5.08 158.80 165.22 171.70 1930 1.00
## y_rep[36] 149.19 0.12 5.12 142.60 149.22 155.70 1966 1.00
## y_rep[37] 158.96 0.13 5.18 152.22 158.98 165.50 1680 1.00
## y_rep[38] 151.24 0.13 5.05 144.58 151.39 157.91 1541 1.00
## y_rep[39] 148.63 0.11 5.22 141.89 148.63 155.24 2254 1.00
## y_rep[40] 152.29 0.12 5.15 145.86 152.30 158.85 1906 1.00
## y_rep[41] 154.21 0.11 5.01 147.80 154.12 160.88 2018 1.00
## y_rep[42] 166.70 0.12 5.02 160.24 166.68 173.07 1654 1.00
## y_rep[43] 156.15 0.11 5.23 149.38 156.10 162.68 2101 1.00
## y_rep[44] 153.94 0.12 5.32 146.89 153.98 160.75 1914 1.00
## y_rep[45] 160.15 0.11 4.98 153.69 160.29 166.48 2004 1.00
## y_rep[46] 163.17 0.11 5.24 156.57 163.20 169.76 2184 1.00
## y_rep[47] 155.42 0.12 5.25 148.67 155.38 162.18 2047 1.00
## y_rep[48] 157.20 0.11 5.22 150.67 157.08 164.15 2108 1.00
## y_rep[49] 161.19 0.12 5.24 154.26 161.07 167.99 1964 1.00
## y_rep[50] 156.85 0.11 5.04 150.44 156.89 163.32 2051 1.00
## y_rep[51] 155.31 0.11 5.06 148.79 155.36 162.02 2007 1.00
## y_rep[52] 152.05 0.11 5.08 145.62 151.92 158.66 2014 1.00
## y_rep[53] 146.84 0.12 5.10 140.35 146.81 153.32 1963 1.00
## y_rep[54] 164.49 0.13 5.16 157.85 164.32 171.18 1659 1.00
## y_rep[55] 147.58 0.11 5.09 141.25 147.61 154.03 2131 1.00
## y_rep[56] 157.77 0.12 5.16 151.28 157.69 164.10 1943 1.00
## y_rep[57] 147.58 0.11 5.09 140.98 147.56 154.14 2035 1.00
## y_rep[58] 156.71 0.11 5.20 149.88 156.70 163.53 2088 1.00
## y_rep[59] 156.67 0.12 5.26 150.04 156.53 163.54 1941 1.00
## y_rep[60] 165.84 0.12 5.13 159.06 165.93 172.49 1930 1.00
## y_rep[61] 148.22 0.11 5.01 141.70 148.36 154.63 1939 1.00
## y_rep[62] 156.51 0.12 5.27 149.87 156.62 163.09 1945 1.00
## y_rep[63] 152.89 0.11 5.17 146.33 152.98 159.30 2034 1.00
## y_rep[64] 159.46 0.11 5.21 152.75 159.44 166.41 2199 1.00
## y_rep[65] 149.74 0.12 5.11 143.32 149.65 156.47 1837 1.00
## y_rep[66] 159.95 0.11 5.08 153.41 159.93 166.63 2024 1.00
## y_rep[67] 149.84 0.12 5.24 143.16 149.79 156.32 1979 1.00
## y_rep[68] 146.88 0.12 5.07 140.42 146.81 153.37 1857 1.00
## y_rep[69] 155.69 0.12 5.11 149.22 155.75 162.28 1963 1.00
## y_rep[70] 151.98 0.12 5.08 145.54 151.88 158.59 1858 1.00
## y_rep[71] 157.11 0.12 5.21 150.60 157.15 163.89 2046 1.00
## y_rep[72] 164.35 0.12 5.23 157.52 164.35 170.98 1839 1.00
## y_rep[73] 157.84 0.12 5.12 151.33 157.92 164.31 1949 1.00
## y_rep[74] 152.66 0.12 5.18 145.83 152.63 159.53 1947 1.00
## y_rep[75] 145.41 0.11 5.09 138.88 145.48 151.85 2159 1.00
## y_rep[76] 165.21 0.11 5.12 158.50 165.33 171.75 2015 1.00
## y_rep[77] 160.52 0.11 4.92 154.28 160.44 166.69 1870 1.00
## y_rep[78] 156.41 0.12 5.20 149.64 156.49 162.88 1938 1.00
## y_rep[79] 150.78 0.11 4.96 144.55 150.83 157.18 1939 1.00
## y_rep[80] 143.78 0.12 5.29 136.98 143.76 150.43 1891 1.00
## y_rep[81] 165.48 0.11 4.99 159.16 165.37 172.09 1969 1.00
## y_rep[82] 152.73 0.12 5.09 146.11 152.88 159.12 1916 1.00
## y_rep[83] 150.09 0.12 5.01 143.68 150.03 156.48 1776 1.00
## y_rep[84] 155.60 0.12 5.18 148.97 155.61 162.01 1994 1.00
## y_rep[85] 150.99 0.12 5.06 144.52 151.00 157.20 1842 1.00
## y_rep[86] 157.18 0.11 5.02 150.73 157.19 163.37 2130 1.00
## y_rep[87] 147.85 0.12 5.13 141.23 147.95 154.38 1788 1.00
## y_rep[88] 154.64 0.12 5.13 148.05 154.63 161.32 1944 1.00
## y_rep[89] 152.22 0.12 5.25 145.52 152.19 158.99 1935 1.00
## y_rep[90] 163.48 0.11 5.00 157.10 163.51 169.79 1987 1.00
## y_rep[91] 162.50 0.11 5.01 156.17 162.67 168.87 1973 1.00
## y_rep[92] 161.13 0.11 5.15 154.58 161.23 167.59 2222 1.00
## y_rep[93] 152.26 0.11 5.05 145.52 152.38 158.57 2100 1.00
## y_rep[94] 158.71 0.11 5.08 152.23 158.86 165.05 2179 1.00
## y_rep[95] 151.73 0.12 5.03 145.11 151.77 158.08 1842 1.00
## y_rep[96] 164.60 0.12 5.12 158.01 164.49 171.32 1949 1.00
## y_rep[97] 152.03 0.12 5.19 145.30 151.97 158.64 2015 1.00
## y_rep[98] 162.11 0.12 5.23 155.38 162.30 168.71 1986 1.00
## y_rep[99] 159.41 0.12 5.15 152.88 159.26 166.06 1979 1.00
## y_rep[100] 152.32 0.11 5.13 145.75 152.40 158.94 2050 1.00
## y_rep[101] 158.48 0.12 4.97 152.17 158.52 164.95 1861 1.00
## y_rep[102] 148.80 0.11 5.19 142.15 148.88 155.62 2248 1.00
## y_rep[103] 158.87 0.12 5.23 151.99 158.89 165.46 1989 1.00
## y_rep[104] 168.06 0.13 5.41 161.22 168.02 174.95 1808 1.00
## y_rep[105] 157.24 0.11 5.04 150.90 157.20 163.67 1920 1.00
## y_rep[106] 149.35 0.11 5.12 142.85 149.33 155.99 2045 1.00
## y_rep[107] 147.36 0.11 5.20 140.61 147.36 154.12 2082 1.00
## y_rep[108] 152.18 0.12 5.15 145.60 152.05 159.04 1928 1.00
## y_rep[109] 154.22 0.11 4.98 147.89 154.06 160.87 1918 1.00
## y_rep[110] 157.13 0.12 5.12 150.48 157.03 163.61 1764 1.00
## y_rep[111] 163.79 0.11 5.07 157.22 163.87 170.34 2014 1.00
## y_rep[112] 143.76 0.11 5.12 137.24 143.61 150.35 2001 1.00
## y_rep[113] 150.76 0.11 5.05 144.44 150.66 157.06 1966 1.00
## y_rep[114] 160.09 0.12 5.00 153.61 160.04 166.49 1783 1.00
## y_rep[115] 158.31 0.12 5.09 151.96 158.39 164.79 1861 1.00
## y_rep[116] 162.36 0.12 5.14 156.14 162.16 168.72 1937 1.00
## y_rep[117] 162.90 0.12 5.12 156.32 162.99 169.37 1899 1.00
## y_rep[118] 164.27 0.11 5.15 157.64 164.37 170.72 2067 1.00
## y_rep[119] 152.15 0.12 5.15 145.58 152.06 158.81 1910 1.00
## y_rep[120] 151.05 0.12 5.08 144.39 151.13 157.66 1925 1.00
## y_rep[121] 158.99 0.11 5.12 152.33 158.97 165.45 2138 1.00
## y_rep[122] 154.19 0.12 5.04 147.88 154.21 160.77 1728 1.00
## y_rep[123] 155.60 0.11 5.00 149.17 155.53 162.07 1981 1.00
## y_rep[124] 151.98 0.11 4.98 145.77 151.91 158.45 2127 1.00
## y_rep[125] 160.24 0.11 5.08 153.87 160.38 166.64 2129 1.00
## y_rep[126] 153.69 0.11 5.18 146.95 153.67 160.30 2101 1.00
## y_rep[127] 142.90 0.12 5.11 136.22 143.03 149.37 1894 1.00
## y_rep[128] 165.34 0.12 5.15 158.92 165.29 171.88 1981 1.00
## y_rep[129] 161.39 0.12 5.31 154.70 161.36 168.07 1847 1.00
## y_rep[130] 147.10 0.11 5.15 140.53 146.98 153.64 2061 1.00
## y_rep[131] 158.13 0.11 5.15 151.57 158.11 164.66 2065 1.00
## y_rep[132] 165.50 0.12 5.15 158.87 165.59 171.97 1865 1.00
## y_rep[133] 151.97 0.11 5.04 145.62 151.99 158.39 1927 1.00
## y_rep[134] 159.36 0.11 4.94 152.92 159.37 165.69 1851 1.00
## y_rep[135] 156.02 0.11 5.08 149.69 155.89 162.62 2087 1.00
## y_rep[136] 169.86 0.12 5.36 162.99 170.00 176.53 1859 1.00
## y_rep[137] 158.20 0.12 5.19 151.67 158.18 164.82 1947 1.00
## y_rep[138] 142.54 0.11 5.17 136.18 142.49 149.07 2022 1.00
## y_rep[139] 145.65 0.12 5.16 138.97 145.57 152.32 1774 1.00
## y_rep[140] 155.17 0.11 4.91 148.99 155.14 161.44 1868 1.00
## y_rep[141] 153.67 0.11 5.11 147.15 153.77 160.09 2034 1.00
## y_rep[142] 155.17 0.11 5.05 148.58 155.12 161.58 2020 1.00
## y_rep[143] 149.24 0.12 4.99 142.79 149.14 155.70 1864 1.00
## y_rep[144] 151.06 0.11 5.03 144.87 151.07 157.37 1975 1.00
## y_rep[145] 154.80 0.11 5.04 148.32 154.76 161.32 1931 1.00
## y_rep[146] 162.38 0.11 5.00 155.80 162.26 169.00 1972 1.00
## y_rep[147] 161.26 0.12 5.22 154.66 161.40 167.83 1981 1.00
## y_rep[148] 149.32 0.12 5.11 142.86 149.28 155.93 1718 1.00
## y_rep[149] 155.22 0.12 5.21 148.40 155.38 161.91 1969 1.00
## y_rep[150] 150.55 0.12 5.10 143.72 150.74 156.86 1957 1.00
## y_rep[151] 160.25 0.12 5.16 153.69 160.26 166.81 1812 1.00
## y_rep[152] 148.79 0.12 5.02 142.56 148.88 155.15 1904 1.00
## y_rep[153] 149.72 0.12 5.26 142.97 149.77 156.34 2006 1.00
## y_rep[154] 154.02 0.12 5.17 147.27 154.02 160.62 2001 1.00
## y_rep[155] 149.76 0.11 5.14 143.22 149.87 156.16 2053 1.00
## y_rep[156] 141.62 0.12 5.45 134.58 141.66 148.54 1949 1.00
## y_rep[157] 156.18 0.11 5.11 149.63 156.11 162.78 2059 1.00
## y_rep[158] 150.49 0.13 5.25 143.47 150.56 157.10 1713 1.00
## y_rep[159] 159.36 0.11 5.12 152.65 159.24 165.74 2183 1.00
## y_rep[160] 151.34 0.11 4.94 145.11 151.39 157.77 1948 1.00
## y_rep[161] 163.45 0.12 5.23 156.57 163.44 170.23 1960 1.00
## y_rep[162] 154.69 0.11 5.13 148.17 154.62 161.18 2087 1.00
## y_rep[163] 146.31 0.12 5.11 139.70 146.32 152.96 1891 1.00
## y_rep[164] 154.81 0.11 5.15 148.14 154.90 161.69 2124 1.00
## y_rep[165] 157.42 0.12 5.17 150.97 157.27 163.96 1894 1.00
## y_rep[166] 155.35 0.12 5.27 148.36 155.38 162.06 1890 1.00
## y_rep[167] 157.70 0.11 5.05 151.24 157.67 164.19 2107 1.00
## y_rep[168] 151.11 0.12 5.22 144.54 150.89 157.79 2039 1.00
## y_rep[169] 148.68 0.11 5.13 142.27 148.58 155.19 2122 1.00
## y_rep[170] 152.02 0.12 4.98 145.51 152.10 158.43 1791 1.00
## y_rep[171] 148.50 0.12 5.14 141.92 148.40 155.01 1883 1.00
## y_rep[172] 157.69 0.12 5.23 151.19 157.69 164.33 1776 1.00
## y_rep[173] 149.10 0.12 5.15 142.53 149.09 155.74 1946 1.00
## y_rep[174] 150.02 0.11 5.14 143.55 149.89 156.60 2046 1.00
## y_rep[175] 159.37 0.11 5.12 152.65 159.44 165.84 2074 1.00
## y_rep[176] 162.96 0.12 5.22 156.35 163.02 169.60 1850 1.00
## y_rep[177] 155.29 0.12 5.20 148.53 155.33 162.13 1801 1.00
## y_rep[178] 149.80 0.12 5.23 143.14 149.97 156.38 1913 1.00
## y_rep[179] 152.87 0.12 5.24 145.94 152.86 159.72 1910 1.00
## y_rep[180] 142.69 0.12 5.21 135.76 142.70 149.20 2023 1.00
## y_rep[181] 155.02 0.12 5.15 148.47 154.93 161.56 1913 1.00
## y_rep[182] 148.22 0.12 5.00 141.81 148.23 154.58 1699 1.00
## y_rep[183] 146.39 0.12 5.13 139.78 146.39 152.95 1924 1.00
## y_rep[184] 150.84 0.12 5.09 144.26 150.80 157.42 1851 1.00
## y_rep[185] 157.21 0.11 5.10 150.53 157.31 163.27 1981 1.00
## y_rep[186] 157.01 0.12 5.04 150.63 156.92 163.58 1799 1.00
## y_rep[187] 155.76 0.11 5.07 149.32 155.57 162.58 2035 1.00
## y_rep[188] 151.14 0.11 5.16 144.50 151.12 157.79 2035 1.00
## y_rep[189] 146.94 0.12 5.18 140.40 146.92 153.73 1955 1.00
## y_rep[190] 157.24 0.11 5.09 150.54 157.30 163.50 2033 1.00
## y_rep[191] 146.95 0.12 5.11 140.43 147.02 153.46 1880 1.00
## y_rep[192] 165.00 0.12 5.17 158.38 165.03 171.73 1999 1.00
## y_rep[193] 148.75 0.12 5.22 142.12 148.66 155.61 1945 1.00
## y_rep[194] 156.90 0.12 5.18 150.24 156.85 163.48 1879 1.00
## y_rep[195] 146.65 0.11 5.05 140.12 146.64 153.24 1954 1.00
## y_rep[196] 151.32 0.11 5.13 144.81 151.43 158.01 2124 1.00
## y_rep[197] 154.38 0.12 5.19 147.70 154.51 161.19 1910 1.00
## y_rep[198] 159.46 0.12 5.21 152.90 159.44 166.14 1917 1.00
## y_rep[199] 164.02 0.12 5.21 157.12 163.98 170.85 1952 1.00
## y_rep[200] 148.11 0.11 5.00 141.63 148.18 154.37 2064 1.00
## y_rep[201] 166.73 0.12 5.31 159.93 166.74 173.62 1933 1.00
## y_rep[202] 154.04 0.12 5.21 147.34 154.17 160.70 1938 1.00
## y_rep[203] 153.97 0.12 5.29 147.06 154.08 160.71 1874 1.00
## y_rep[204] 154.21 0.11 5.06 147.96 154.22 160.72 1973 1.00
## y_rep[205] 154.05 0.12 5.11 147.54 153.96 160.67 1855 1.00
## y_rep[206] 150.28 0.11 5.03 143.80 150.33 156.71 1960 1.00
## y_rep[207] 153.87 0.12 5.03 147.27 153.87 160.41 1827 1.00
## y_rep[208] 145.06 0.11 5.13 138.42 145.08 151.60 2004 1.00
## y_rep[209] 143.50 0.12 5.16 136.83 143.44 150.20 2007 1.00
## y_rep[210] 146.43 0.12 5.12 139.89 146.40 152.89 1851 1.00
## y_rep[211] 158.72 0.12 5.12 152.08 158.93 165.26 1963 1.00
## y_rep[212] 162.17 0.12 5.07 155.91 162.10 168.68 1903 1.00
## y_rep[213] 147.61 0.12 5.05 141.17 147.61 154.02 1892 1.00
## y_rep[214] 150.69 0.11 5.08 143.96 150.79 157.01 2004 1.00
## y_rep[215] 159.78 0.12 5.09 153.42 159.74 166.30 1752 1.00
## y_rep[216] 153.71 0.12 5.26 147.03 153.76 160.37 1980 1.00
## y_rep[217] 159.19 0.12 5.16 152.38 159.36 165.75 2009 1.00
## y_rep[218] 144.38 0.11 5.21 137.67 144.34 150.97 2245 1.00
## y_rep[219] 153.59 0.11 5.06 147.12 153.46 160.00 1984 1.00
## y_rep[220] 157.53 0.12 5.24 151.41 157.38 164.24 1889 1.00
## y_rep[221] 152.42 0.11 5.06 145.88 152.39 159.20 2063 1.00
## y_rep[222] 144.50 0.12 5.21 137.76 144.54 151.16 1932 1.00
## y_rep[223] 150.00 0.12 5.12 143.41 149.90 156.45 1874 1.00
## y_rep[224] 146.53 0.11 5.15 139.95 146.66 153.10 2146 1.00
## y_rep[225] 160.08 0.12 5.30 153.27 160.20 166.81 1842 1.00
## y_rep[226] 155.14 0.12 5.08 148.56 155.30 161.48 1920 1.00
## y_rep[227] 158.45 0.12 4.98 152.02 158.63 164.86 1870 1.00
## y_rep[228] 167.41 0.12 5.16 160.98 167.46 174.02 1975 1.00
## y_rep[229] 153.09 0.12 5.08 146.62 153.10 159.58 1660 1.00
## y_rep[230] 161.01 0.12 5.09 154.56 161.01 167.59 1868 1.00
## y_rep[231] 159.69 0.12 5.18 153.13 159.58 166.53 1999 1.00
## y_rep[232] 145.00 0.12 5.19 138.30 145.01 151.71 1990 1.00
## y_rep[233] 149.50 0.12 5.18 142.79 149.53 156.23 1979 1.00
## y_rep[234] 167.78 0.12 5.12 161.17 167.82 174.48 1946 1.00
## y_rep[235] 161.02 0.11 5.08 154.63 161.07 167.68 2077 1.00
## y_rep[236] 150.06 0.11 4.97 143.81 150.11 156.18 2048 1.00
## y_rep[237] 153.79 0.12 5.09 147.26 153.82 160.34 1884 1.00
## y_rep[238] 163.24 0.13 5.16 156.64 163.17 169.65 1642 1.00
## y_rep[239] 154.77 0.11 5.10 148.40 154.63 161.46 2017 1.00
## y_rep[240] 151.55 0.12 5.24 144.96 151.47 158.03 1887 1.00
## y_rep[241] 151.82 0.13 5.21 145.17 151.86 158.42 1685 1.00
## y_rep[242] 152.32 0.12 5.09 145.84 152.46 158.71 1892 1.00
## y_rep[243] 152.83 0.13 5.12 146.24 152.80 159.56 1627 1.00
## y_rep[244] 150.97 0.12 5.18 144.19 151.04 157.45 1959 1.00
## y_rep[245] 152.71 0.12 4.97 146.50 152.65 158.92 1834 1.00
## y_rep[246] 155.76 0.12 5.10 148.99 155.81 162.31 1843 1.00
## y_rep[247] 157.26 0.12 5.14 150.64 157.35 163.61 1819 1.00
## y_rep[248] 152.47 0.11 5.14 145.96 152.49 159.03 2071 1.00
## y_rep[249] 157.73 0.11 5.00 151.40 157.74 164.08 1943 1.00
## y_rep[250] 155.42 0.11 5.07 149.04 155.46 161.94 2094 1.00
## y_rep[251] 161.78 0.11 5.18 155.14 161.85 168.31 2104 1.00
## y_rep[252] 157.03 0.12 5.25 150.23 157.06 163.62 2031 1.00
## y_rep[253] 153.18 0.12 5.25 146.57 153.14 159.93 1920 1.00
## y_rep[254] 153.33 0.12 5.07 146.89 153.40 159.83 1858 1.00
## y_rep[255] 162.05 0.12 5.20 155.52 162.04 168.69 1738 1.00
## y_rep[256] 154.28 0.12 5.09 147.86 154.31 160.73 1803 1.00
## y_rep[257] 156.56 0.11 5.10 150.09 156.58 163.11 2062 1.00
## y_rep[258] 142.44 0.12 5.19 135.97 142.41 149.15 1883 1.00
## y_rep[259] 165.06 0.12 5.23 158.56 164.97 171.71 1878 1.00
## y_rep[260] 164.50 0.12 5.10 158.02 164.54 170.88 1846 1.00
## y_rep[261] 156.92 0.11 5.12 150.61 156.95 163.46 1995 1.00
## y_rep[262] 154.90 0.12 5.14 148.16 155.04 161.35 1837 1.00
## y_rep[263] 155.07 0.12 5.22 148.31 155.01 161.70 1925 1.00
## y_rep[264] 158.27 0.13 5.09 151.82 158.22 164.90 1596 1.00
## y_rep[265] 155.92 0.11 5.09 149.43 156.01 162.29 1968 1.00
## y_rep[266] 153.69 0.11 5.00 147.11 153.86 160.06 1931 1.00
## y_rep[267] 164.85 0.12 5.08 158.25 164.86 171.54 1937 1.00
## y_rep[268] 150.20 0.12 5.11 143.72 150.17 156.70 1934 1.00
## y_rep[269] 159.91 0.11 5.12 153.38 159.92 166.52 2022 1.00
## y_rep[270] 155.71 0.12 5.26 148.93 155.65 162.30 1961 1.00
## y_rep[271] 158.59 0.12 5.15 152.08 158.52 165.13 1979 1.00
## y_rep[272] 153.70 0.11 5.10 147.04 153.75 160.27 1974 1.00
## y_rep[273] 159.89 0.11 5.09 153.16 159.92 166.49 2046 1.00
## y_rep[274] 150.74 0.12 5.32 144.19 150.65 157.50 1857 1.00
## y_rep[275] 147.71 0.12 5.14 141.26 147.51 154.15 1892 1.00
## y_rep[276] 160.44 0.11 5.28 153.63 160.58 167.12 2240 1.00
## y_rep[277] 154.37 0.11 5.08 147.95 154.38 161.02 2065 1.00
## y_rep[278] 153.27 0.11 5.10 146.71 153.42 159.74 1977 1.00
## y_rep[279] 156.21 0.13 5.22 149.36 156.34 162.69 1730 1.00
## y_rep[280] 149.75 0.12 5.10 143.08 149.79 156.29 1912 1.00
## y_rep[281] 145.28 0.12 5.25 138.45 145.34 151.91 1945 1.00
## y_rep[282] 149.39 0.12 5.10 142.52 149.52 155.71 1811 1.00
## y_rep[283] 147.78 0.12 5.10 141.12 147.87 154.29 1825 1.00
## y_rep[284] 157.84 0.12 5.07 151.50 157.84 164.32 1788 1.00
## y_rep[285] 146.06 0.12 5.16 139.37 146.12 152.64 1867 1.00
## y_rep[286] 153.66 0.11 4.96 147.50 153.75 160.09 1935 1.00
## y_rep[287] 157.48 0.12 5.30 150.71 157.44 164.01 1831 1.00
## y_rep[288] 159.89 0.11 5.15 153.31 159.85 166.48 2127 1.00
## y_rep[289] 153.78 0.12 5.08 147.11 153.95 160.19 1702 1.00
## y_rep[290] 157.33 0.11 5.07 151.10 157.25 163.81 2047 1.00
## y_rep[291] 156.10 0.12 5.14 149.51 156.18 162.61 1780 1.00
## y_rep[292] 165.15 0.12 5.23 158.44 165.30 171.81 2026 1.00
## y_rep[293] 157.86 0.12 5.16 151.11 157.84 164.35 1927 1.00
## y_rep[294] 157.47 0.11 5.23 150.69 157.51 164.09 2119 1.00
## y_rep[295] 156.93 0.11 5.17 150.41 156.95 163.47 2053 1.00
## y_rep[296] 145.90 0.12 5.17 139.46 145.89 152.56 1867 1.00
## y_rep[297] 147.04 0.12 5.09 140.49 147.01 153.48 1911 1.00
## y_rep[298] 158.16 0.12 5.19 151.54 158.11 164.77 1755 1.00
## y_rep[299] 160.06 0.12 5.10 153.47 160.05 166.55 1886 1.00
## y_rep[300] 155.99 0.12 5.23 149.33 156.11 162.72 1805 1.00
## y_rep[301] 147.46 0.12 5.12 140.87 147.44 154.16 1943 1.00
## y_rep[302] 157.82 0.11 5.07 151.43 157.75 164.37 1967 1.00
## y_rep[303] 148.35 0.12 5.16 141.84 148.41 154.80 1772 1.00
## y_rep[304] 156.63 0.11 5.05 149.91 156.80 163.02 2094 1.00
## y_rep[305] 164.00 0.12 5.19 157.09 164.21 170.65 1888 1.00
## y_rep[306] 162.85 0.12 5.15 156.46 162.83 169.47 1882 1.00
## y_rep[307] 157.76 0.11 5.08 151.27 157.81 164.32 2008 1.00
## y_rep[308] 153.07 0.11 5.07 146.56 153.16 159.50 2048 1.00
## y_rep[309] 144.91 0.12 5.27 137.93 144.90 151.66 2028 1.00
## y_rep[310] 159.16 0.12 5.06 152.98 159.12 165.72 1923 1.00
## y_rep[311] 154.01 0.12 5.15 147.31 154.12 160.40 1941 1.00
## y_rep[312] 144.26 0.12 5.26 137.44 144.35 150.97 1943 1.00
## y_rep[313] 156.55 0.12 5.14 150.01 156.54 163.18 1942 1.00
## y_rep[314] 155.78 0.12 5.20 149.21 155.82 162.38 1972 1.00
## y_rep[315] 156.71 0.12 5.09 150.12 156.54 163.52 1881 1.00
## y_rep[316] 150.95 0.12 5.06 144.60 150.94 157.37 1780 1.00
## y_rep[317] 159.43 0.11 5.12 152.60 159.51 165.99 2063 1.00
## y_rep[318] 154.88 0.11 5.13 148.18 154.80 161.44 2111 1.00
## y_rep[319] 152.19 0.12 5.14 145.74 152.08 158.73 1858 1.00
## y_rep[320] 151.56 0.12 5.07 145.13 151.45 158.20 1849 1.00
## y_rep[321] 154.85 0.12 5.06 148.40 154.81 161.12 1931 1.00
## y_rep[322] 160.24 0.12 5.26 153.38 160.31 167.00 1801 1.00
## y_rep[323] 158.47 0.11 5.00 152.29 158.44 164.63 2205 1.00
## y_rep[324] 160.48 0.12 5.11 154.06 160.42 166.93 1961 1.00
## y_rep[325] 153.74 0.12 5.04 147.21 153.59 160.28 1848 1.00
## y_rep[326] 146.01 0.12 5.19 139.36 145.97 152.72 1994 1.00
## y_rep[327] 152.55 0.12 5.28 145.79 152.51 159.33 1898 1.00
## y_rep[328] 156.04 0.12 5.18 149.43 156.13 162.64 1887 1.00
## y_rep[329] 151.81 0.11 5.10 145.34 151.79 158.36 2001 1.00
## y_rep[330] 148.39 0.12 5.11 142.08 148.35 154.87 1888 1.00
## y_rep[331] 153.12 0.11 5.09 146.54 153.16 159.56 1992 1.00
## y_rep[332] 162.52 0.11 5.14 155.96 162.34 169.07 2015 1.00
## y_rep[333] 152.68 0.12 5.18 146.20 152.49 159.48 2012 1.00
## y_rep[334] 149.86 0.12 5.24 143.08 149.80 156.72 2065 1.00
## y_rep[335] 153.08 0.11 5.15 146.66 153.09 159.80 2177 1.00
## y_rep[336] 161.73 0.12 5.17 155.06 161.81 168.52 2013 1.00
## y_rep[337] 148.72 0.11 5.19 141.93 148.71 155.50 2043 1.00
## y_rep[338] 152.55 0.12 5.15 145.86 152.47 159.30 1974 1.00
## y_rep[339] 146.42 0.12 5.12 140.00 146.37 153.13 1957 1.00
## y_rep[340] 153.75 0.12 5.10 147.24 153.76 160.24 1701 1.00
## y_rep[341] 148.36 0.12 5.16 141.65 148.42 154.88 1929 1.00
## y_rep[342] 153.73 0.11 5.06 147.15 153.60 160.46 2086 1.00
## y_rep[343] 153.93 0.12 5.16 147.34 154.05 160.24 1891 1.00
## y_rep[344] 157.05 0.12 5.16 150.74 156.88 163.76 1953 1.00
## y_rep[345] 149.43 0.13 5.10 142.96 149.39 155.93 1478 1.00
## y_rep[346] 151.09 0.12 5.09 144.49 151.19 157.41 1913 1.00
## y_rep[347] 150.99 0.12 5.19 144.33 151.10 157.49 1969 1.00
## y_rep[348] 156.47 0.12 5.19 149.84 156.32 163.32 2033 1.00
## y_rep[349] 144.67 0.12 5.15 138.25 144.59 151.43 1946 1.00
## y_rep[350] 161.04 0.12 5.27 154.58 161.01 167.98 2046 1.00
## y_rep[351] 162.79 0.12 5.02 156.18 162.83 169.26 1644 1.00
## y_rep[352] 161.62 0.11 5.00 155.19 161.75 167.96 2111 1.00
## lp__ -749.57 0.06 1.45 -751.51 -749.24 -748.08 564 1.00
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:57:32 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).summary(lm(height~weight.z+I(weight.z^2),data=howell2))##
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.7517 -2.8933 0.0313 3.1419 14.7936
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 154.60953 0.35209 439.120 <2e-16 ***
## weight.z 5.84515 0.27348 21.373 <2e-16 ***
## I(weight.z^2) -0.01248 0.22483 -0.055 0.956
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.094 on 349 degrees of freedom
## Multiple R-squared: 0.5696, Adjusted R-squared: 0.5672
## F-statistic: 231 on 2 and 349 DF, p-value: < 2.2e-16See Figure 4.9 of Statistical rethinking
https://github.com/ssp3nc3r/rethinking/blob/master/chapter04.Rmd
Here, all the variance is represented (variance of actual heights).
5.2 Howell 1: data set with all individuals
Standardizing the explanatory variable:
howell1 <- howell1 %>% mutate(weight.z = (weight - mean(weight)) / sd(weight))Data in a list:
m7.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z)data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
}
parameters { // unobserved variables
real beta[3]; // parameters: intercept and slope
real sigma;
}
model {
height ~ normal(beta[1] + beta[2]*weight + beta[3]*(weight .* weight),sigma); // likelihood
beta[1] ~ normal(178, 100); // prior of the intercept
beta[2] ~ normal(0, 100); // prior of the slope
beta[3] ~ normal(0, 100);
sigma ~ uniform(0,50);
}
generated quantities {
vector[N] y_rep;
for(i in 1:N)
y_rep[i] = normal_rng(beta[1] + beta[2]*weight[i] + beta[3]*(weight[i] .* weight[i]), sigma);
}fit.m7 <- sampling(m7, data = m7.data, iter = 1000, chains = 4, cores = 4)
print(fit.m7, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: c7fbeba14f23c52987bbbb1ee94a3a29.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 146.65 0.01 0.36 146.19 146.64 147.12 901 1.00
## beta[2] 21.42 0.01 0.27 21.06 21.42 21.76 597 1.01
## beta[3] -8.40 0.01 0.28 -8.77 -8.40 -8.05 712 1.00
## sigma 5.77 0.00 0.18 5.55 5.77 6.00 2401 1.00
## y_rep[1] 158.78 0.13 5.73 151.57 158.81 165.97 1978 1.00
## y_rep[2] 148.02 0.13 5.79 140.74 148.02 155.29 1963 1.00
## y_rep[3] 140.64 0.13 5.92 132.97 140.75 148.15 1961 1.00
## y_rep[4] 160.20 0.13 5.68 152.71 160.24 167.62 2047 1.00
## y_rep[5] 153.52 0.14 5.81 146.05 153.51 161.07 1824 1.00
## y_rep[6] 157.46 0.14 5.93 149.74 157.42 165.19 1895 1.00
## y_rep[7] 150.15 0.13 5.82 142.68 150.33 157.44 1893 1.00
## y_rep[8] 160.32 0.12 5.65 153.27 160.21 167.48 2076 1.00
## y_rep[9] 145.63 0.13 5.65 138.16 145.71 152.86 1864 1.00
## y_rep[10] 160.21 0.13 5.61 152.95 160.16 167.56 1998 1.00
## y_rep[11] 159.64 0.13 5.70 152.26 159.59 166.98 2070 1.00
## y_rep[12] 153.49 0.13 5.83 146.06 153.42 161.07 1994 1.00
## y_rep[13] 147.28 0.13 5.76 140.05 147.25 154.60 1968 1.00
## y_rep[14] 158.59 0.13 5.77 151.28 158.67 166.10 2035 1.00
## y_rep[15] 143.92 0.13 5.87 136.10 144.00 151.52 2123 1.00
## y_rep[16] 159.01 0.13 5.76 151.62 159.16 166.42 1859 1.00
## y_rep[17] 154.47 0.13 5.73 146.96 154.41 161.77 1963 1.00
## y_rep[18] 150.45 0.14 5.88 142.95 150.52 158.11 1895 1.00
## y_rep[19] 113.36 0.13 5.80 106.08 113.26 120.78 2021 1.00
## y_rep[20] 97.13 0.13 5.69 89.84 97.06 104.28 1909 1.00
## y_rep[21] 85.82 0.14 5.84 78.51 85.82 93.26 1764 1.00
## y_rep[22] 159.14 0.13 5.79 151.87 159.07 166.45 1953 1.00
## y_rep[23] 154.85 0.12 5.69 147.69 154.86 162.04 2209 1.00
## y_rep[24] 123.54 0.13 5.86 115.90 123.55 131.15 1939 1.00
## y_rep[25] 103.07 0.13 5.85 95.76 102.94 110.63 1940 1.00
## y_rep[26] 146.68 0.13 5.83 139.39 146.73 154.21 1974 1.00
## y_rep[27] 149.77 0.13 5.72 142.39 149.93 157.14 1906 1.00
## y_rep[28] 146.43 0.15 5.74 139.19 146.42 153.73 1467 1.00
## y_rep[29] 132.13 0.13 5.72 124.58 131.99 139.65 1885 1.00
## y_rep[30] 121.27 0.13 5.69 113.88 121.09 128.85 2004 1.00
## y_rep[31] 108.58 0.14 6.01 100.86 108.55 116.59 1832 1.00
## y_rep[32] 152.70 0.13 5.78 145.23 152.76 160.18 1956 1.00
## y_rep[33] 160.13 0.14 5.90 152.56 160.08 167.75 1910 1.00
## y_rep[34] 149.10 0.13 5.90 141.52 149.12 156.69 2081 1.00
## y_rep[35] 150.44 0.13 5.71 143.18 150.50 157.77 1825 1.00
## y_rep[36] 157.98 0.13 5.71 150.48 158.06 165.33 2063 1.00
## y_rep[37] 150.80 0.13 5.71 143.32 150.87 157.98 1990 1.00
## y_rep[38] 146.49 0.12 5.62 139.17 146.54 153.76 2122 1.00
## y_rep[39] 158.26 0.12 5.58 151.17 158.17 165.51 2208 1.00
## y_rep[40] 158.59 0.14 5.90 150.82 158.59 166.19 1658 1.00
## y_rep[41] 159.19 0.13 5.84 151.85 159.08 166.86 2033 1.00
## y_rep[42] 159.30 0.13 5.85 152.05 159.14 166.92 1955 1.00
## y_rep[43] 159.94 0.14 5.84 152.36 160.13 167.27 1666 1.00
## y_rep[44] 151.32 0.14 5.79 143.76 151.42 158.56 1792 1.00
## y_rep[45] 159.50 0.13 5.84 151.84 159.41 166.91 1932 1.00
## y_rep[46] 134.91 0.14 5.73 127.64 134.80 142.25 1714 1.00
## y_rep[47] 115.53 0.14 5.88 107.94 115.45 123.09 1675 1.00
## y_rep[48] 123.05 0.12 5.69 116.07 122.80 130.57 2220 1.00
## y_rep[49] 94.71 0.13 5.67 87.59 94.79 101.95 1891 1.00
## y_rep[50] 153.53 0.13 5.88 146.04 153.63 160.99 1898 1.00
## y_rep[51] 150.57 0.12 5.68 142.99 150.59 157.89 2156 1.00
## y_rep[52] 155.16 0.13 5.75 147.73 155.23 162.57 2107 1.00
## y_rep[53] 156.76 0.12 5.76 149.39 156.76 163.96 2127 1.00
## y_rep[54] 119.70 0.13 5.81 112.29 119.67 127.17 1869 1.00
## y_rep[55] 101.51 0.13 5.75 94.36 101.55 108.71 1898 1.00
## y_rep[56] 93.14 0.14 5.85 85.88 92.96 100.97 1739 1.00
## y_rep[57] 159.52 0.13 5.80 152.12 159.55 166.88 2096 1.00
## y_rep[58] 157.98 0.13 5.74 150.60 158.03 165.43 1881 1.00
## y_rep[59] 156.39 0.13 6.03 148.74 156.42 164.20 2000 1.00
## y_rep[60] 159.88 0.14 5.85 152.43 159.81 167.55 1798 1.00
## y_rep[61] 160.26 0.13 5.82 152.68 160.41 167.81 1977 1.00
## y_rep[62] 157.50 0.13 5.81 150.12 157.52 164.88 2147 1.00
## y_rep[63] 158.78 0.13 5.64 151.62 158.81 165.96 1938 1.00
## y_rep[64] 160.09 0.13 5.86 152.57 160.09 167.56 1953 1.00
## y_rep[65] 158.60 0.14 5.87 151.08 158.53 166.31 1655 1.00
## y_rep[66] 157.30 0.13 5.76 149.83 157.26 164.64 2004 1.00
## y_rep[67] 154.73 0.13 5.73 147.41 154.74 162.00 1908 1.00
## y_rep[68] 147.92 0.13 5.54 140.88 147.93 155.04 1836 1.00
## y_rep[69] 160.15 0.13 5.89 152.72 160.03 167.57 1993 1.00
## y_rep[70] 148.64 0.13 5.89 140.93 148.77 155.95 2129 1.01
## y_rep[71] 125.74 0.13 5.81 118.20 125.83 133.26 1944 1.00
## y_rep[72] 128.25 0.13 5.68 120.82 128.31 135.51 2009 1.00
## y_rep[73] 159.10 0.13 5.67 151.91 159.05 166.39 1884 1.00
## y_rep[74] 148.92 0.13 5.73 141.44 149.01 156.25 1969 1.00
## y_rep[75] 136.57 0.13 5.89 128.85 136.76 144.09 1973 1.00
## y_rep[76] 158.33 0.13 5.81 150.93 158.20 165.71 2127 1.00
## y_rep[77] 158.29 0.13 5.74 150.93 158.23 165.73 2082 1.00
## y_rep[78] 93.78 0.13 5.55 86.65 93.73 100.95 1826 1.00
## y_rep[79] 160.02 0.13 5.77 152.63 160.05 167.34 1861 1.00
## y_rep[80] 149.81 0.14 5.86 142.20 149.83 157.30 1825 1.00
## y_rep[81] 154.21 0.13 5.71 146.77 154.28 161.70 1873 1.00
## y_rep[82] 132.38 0.13 5.70 125.07 132.44 139.57 1941 1.00
## y_rep[83] 158.36 0.13 5.72 151.10 158.18 165.77 1976 1.00
## y_rep[84] 155.65 0.13 5.71 148.27 155.67 162.90 2015 1.00
## y_rep[85] 159.66 0.13 5.69 152.40 159.85 166.85 1908 1.00
## y_rep[86] 151.48 0.14 5.94 143.92 151.41 159.03 1908 1.00
## y_rep[87] 107.45 0.13 5.94 99.80 107.41 114.98 2179 1.00
## y_rep[88] 159.87 0.13 5.83 152.30 159.81 167.35 1980 1.00
## y_rep[89] 151.63 0.13 5.83 144.17 151.68 158.83 1996 1.00
## y_rep[90] 147.51 0.14 5.89 139.89 147.57 154.91 1890 1.00
## y_rep[91] 157.89 0.14 5.77 150.29 158.05 165.22 1760 1.00
## y_rep[92] 154.74 0.13 5.77 147.31 154.75 161.82 2015 1.00
## y_rep[93] 158.56 0.13 5.88 150.97 158.53 166.24 2008 1.00
## y_rep[94] 160.27 0.13 5.74 152.78 160.35 167.40 1852 1.00
## y_rep[95] 111.24 0.13 5.83 103.59 111.22 118.70 2083 1.00
## y_rep[96] 158.93 0.13 5.82 151.41 159.07 166.32 2157 1.00
## y_rep[97] 155.14 0.13 5.76 147.50 155.16 162.68 1941 1.00
## y_rep[98] 145.92 0.13 5.98 138.14 145.80 153.54 2069 1.00
## y_rep[99] 108.88 0.13 5.93 101.10 108.88 116.48 2120 1.00
## y_rep[100] 160.00 0.14 5.75 152.89 159.80 167.50 1674 1.00
## y_rep[101] 132.25 0.13 5.90 124.58 132.17 139.77 1927 1.00
## y_rep[102] 159.91 0.13 5.82 152.53 160.00 167.34 2141 1.00
## y_rep[103] 158.32 0.14 5.95 151.10 158.14 166.02 1919 1.00
## y_rep[104] 153.39 0.13 5.96 145.77 153.37 161.02 2099 1.00
## y_rep[105] 141.92 0.13 5.80 134.43 142.01 149.44 1978 1.00
## y_rep[106] 160.14 0.14 5.88 152.63 160.06 167.75 1870 1.00
## y_rep[107] 155.55 0.13 5.66 148.37 155.52 162.67 2008 1.00
## y_rep[108] 152.31 0.13 5.85 144.82 152.41 159.60 2045 1.00
## y_rep[109] 141.61 0.12 5.74 134.42 141.56 149.02 2201 1.00
## y_rep[110] 106.44 0.15 5.83 99.16 106.40 113.88 1443 1.00
## y_rep[111] 104.74 0.13 5.98 97.22 104.62 112.49 1983 1.00
## y_rep[112] 157.36 0.14 5.77 149.95 157.28 164.62 1677 1.00
## y_rep[113] 153.48 0.13 5.79 146.21 153.56 160.65 1953 1.00
## y_rep[114] 109.67 0.13 5.75 102.10 109.75 117.06 1884 1.00
## y_rep[115] 100.59 0.13 5.73 93.19 100.49 107.79 2038 1.00
## y_rep[116] 88.95 0.13 5.97 81.38 88.89 96.53 1975 1.00
## y_rep[117] 158.36 0.14 5.78 150.97 158.42 165.57 1826 1.00
## y_rep[118] 149.15 0.14 5.96 141.55 149.15 156.85 1895 1.00
## y_rep[119] 156.77 0.13 5.72 149.34 156.77 163.98 1839 1.00
## y_rep[120] 154.84 0.12 5.76 147.41 154.66 162.29 2156 1.00
## y_rep[121] 160.36 0.14 5.98 152.64 160.26 168.07 1916 1.00
## y_rep[122] 160.29 0.14 5.80 152.75 160.16 167.73 1812 1.00
## y_rep[123] 160.05 0.14 5.80 152.68 160.07 167.45 1753 1.00
## y_rep[124] 83.29 0.13 5.80 76.02 83.20 90.77 1998 1.00
## y_rep[125] 154.74 0.13 5.81 147.31 154.86 162.01 2130 1.00
## y_rep[126] 159.68 0.13 5.83 152.04 159.53 167.48 2057 1.00
## y_rep[127] 94.34 0.13 5.84 86.95 94.43 101.73 1959 1.00
## y_rep[128] 153.88 0.14 5.94 146.24 153.95 161.54 1719 1.00
## y_rep[129] 160.00 0.13 5.83 152.73 159.80 167.62 1924 1.00
## y_rep[130] 154.46 0.13 5.99 146.61 154.36 161.97 2098 1.00
## y_rep[131] 112.89 0.13 5.88 105.45 112.69 120.58 1899 1.00
## y_rep[132] 100.40 0.13 5.95 93.05 100.24 108.02 2073 1.00
## y_rep[133] 160.14 0.13 5.83 152.38 160.17 167.78 1997 1.00
## y_rep[134] 159.72 0.13 5.79 152.29 159.79 167.01 1910 1.00
## y_rep[135] 154.96 0.14 5.80 147.64 154.90 162.40 1788 1.00
## y_rep[136] 159.26 0.13 5.84 151.76 159.31 166.80 1915 1.00
## y_rep[137] 150.71 0.13 5.63 143.43 150.61 157.88 1807 1.00
## y_rep[138] 159.54 0.13 5.77 152.25 159.46 166.93 2019 1.00
## y_rep[139] 135.98 0.13 5.67 128.62 136.04 143.18 1986 1.00
## y_rep[140] 159.16 0.13 6.00 151.40 159.17 166.82 2076 1.00
## y_rep[141] 89.61 0.13 5.74 82.30 89.52 97.23 1899 1.00
## y_rep[142] 158.84 0.13 5.90 151.40 158.81 166.51 2004 1.00
## y_rep[143] 151.48 0.13 5.79 144.29 151.49 159.00 2007 1.00
## y_rep[144] 150.24 0.13 5.67 142.76 150.18 157.50 1946 1.00
## y_rep[145] 119.78 0.13 5.84 112.64 119.67 127.19 2058 1.00
## y_rep[146] 148.49 0.13 5.80 140.96 148.50 155.78 1904 1.00
## y_rep[147] 154.66 0.13 5.80 147.18 154.82 161.98 1880 1.00
## y_rep[148] 113.98 0.13 5.79 106.49 114.06 121.26 1955 1.00
## y_rep[149] 105.47 0.13 5.78 98.21 105.28 113.13 1959 1.00
## y_rep[150] 156.60 0.13 5.83 149.21 156.54 164.17 1907 1.00
## y_rep[151] 158.66 0.13 5.62 151.40 158.70 165.63 1958 1.00
## y_rep[152] 117.42 0.14 5.87 109.80 117.29 124.84 1767 1.00
## y_rep[153] 111.78 0.14 5.79 104.54 111.71 119.27 1777 1.00
## y_rep[154] 160.33 0.13 5.75 152.79 160.46 167.55 1936 1.00
## y_rep[155] 142.49 0.13 5.81 135.30 142.26 150.18 2029 1.00
## y_rep[156] 153.34 0.13 5.83 146.00 153.34 160.89 1926 1.00
## y_rep[157] 94.96 0.14 5.88 87.45 94.88 102.52 1828 1.00
## y_rep[158] 159.67 0.13 5.67 152.31 159.68 166.99 1979 1.00
## y_rep[159] 159.11 0.13 5.70 151.83 158.94 166.45 1963 1.00
## y_rep[160] 160.38 0.13 5.80 153.27 160.18 167.82 1886 1.00
## y_rep[161] 160.27 0.13 5.78 152.76 160.19 167.73 1971 1.00
## y_rep[162] 160.37 0.14 5.85 153.02 160.32 167.96 1786 1.00
## y_rep[163] 154.59 0.13 5.89 147.32 154.55 162.05 1947 1.00
## y_rep[164] 120.46 0.13 5.82 113.08 120.44 127.97 2078 1.00
## y_rep[165] 153.20 0.13 5.78 145.59 153.31 160.32 2077 1.00
## y_rep[166] 159.51 0.13 5.77 152.04 159.60 166.65 2011 1.00
## y_rep[167] 156.29 0.13 5.71 148.94 156.21 163.67 2002 1.00
## y_rep[168] 157.38 0.13 5.75 150.04 157.21 164.72 1937 1.00
## y_rep[169] 154.40 0.13 5.81 147.13 154.35 161.75 1882 1.00
## y_rep[170] 159.93 0.13 5.71 152.59 159.99 167.30 2024 1.00
## y_rep[171] 93.16 0.14 5.86 85.96 93.04 100.96 1784 1.00
## y_rep[172] 93.90 0.13 5.83 86.33 93.89 101.29 2161 1.00
## y_rep[173] 156.20 0.13 5.66 148.91 156.19 163.44 1934 1.00
## y_rep[174] 141.43 0.13 5.85 134.07 141.42 149.14 1985 1.00
## y_rep[175] 159.89 0.13 5.81 152.38 159.82 167.30 2053 1.00
## y_rep[176] 160.19 0.13 5.84 152.63 160.13 167.83 1907 1.00
## y_rep[177] 147.77 0.13 5.89 140.21 147.81 155.31 1916 1.00
## y_rep[178] 159.07 0.14 5.90 151.41 158.84 166.62 1865 1.00
## y_rep[179] 160.13 0.15 5.92 152.44 160.10 167.78 1627 1.00
## y_rep[180] 154.30 0.13 5.77 146.87 154.37 161.76 1990 1.00
## y_rep[181] 159.59 0.13 5.80 152.01 159.63 167.15 1997 1.00
## y_rep[182] 157.86 0.13 5.82 150.51 157.96 165.49 1922 1.00
## y_rep[183] 158.35 0.14 5.90 150.76 158.46 165.89 1823 1.00
## y_rep[184] 159.17 0.14 5.82 151.50 159.20 166.87 1721 1.00
## y_rep[185] 140.16 0.12 5.94 132.72 140.06 147.74 2271 1.00
## y_rep[186] 141.20 0.13 5.70 133.88 141.32 148.45 1785 1.00
## y_rep[187] 123.82 0.13 5.80 116.29 123.85 131.33 2035 1.00
## y_rep[188] 125.36 0.14 5.67 118.02 125.24 132.51 1744 1.00
## y_rep[189] 118.57 0.13 5.87 111.17 118.42 126.17 2018 1.00
## y_rep[190] 146.27 0.13 5.85 138.56 146.36 153.60 1931 1.00
## y_rep[191] 157.37 0.13 5.80 149.80 157.44 164.91 1950 1.00
## y_rep[192] 156.01 0.13 5.75 148.68 155.98 163.36 1925 1.00
## y_rep[193] 157.29 0.13 5.64 150.00 157.22 164.50 1946 1.00
## y_rep[194] 151.24 0.13 5.84 143.64 151.21 158.88 1997 1.00
## y_rep[195] 153.88 0.15 5.88 145.98 153.90 161.22 1626 1.00
## y_rep[196] 94.28 0.14 5.85 86.91 94.19 101.80 1851 1.00
## y_rep[197] 157.08 0.13 5.70 149.56 157.13 164.40 1941 1.00
## y_rep[198] 160.28 0.13 5.71 153.25 160.11 167.72 1969 1.00
## y_rep[199] 160.17 0.13 5.71 152.72 160.34 167.49 1953 1.00
## y_rep[200] 110.11 0.13 5.67 102.75 110.07 117.57 2031 1.00
## y_rep[201] 96.19 0.13 5.84 88.72 96.15 103.64 2113 1.00
## y_rep[202] 151.23 0.13 5.83 143.80 151.18 158.87 1944 1.00
## y_rep[203] 92.26 0.13 5.84 84.44 92.45 99.49 1973 1.00
## y_rep[204] 157.62 0.12 5.74 150.28 157.55 164.97 2158 1.00
## y_rep[205] 152.72 0.13 5.75 145.37 152.77 160.02 1872 1.00
## y_rep[206] 160.29 0.14 5.96 152.66 160.48 167.85 1942 1.01
## y_rep[207] 150.81 0.13 5.63 143.67 150.94 158.09 1814 1.00
## y_rep[208] 151.34 0.13 5.75 143.83 151.33 158.64 2089 1.00
## y_rep[209] 156.41 0.13 5.76 149.13 156.27 163.79 2032 1.00
## y_rep[210] 151.68 0.14 5.81 143.99 151.80 158.97 1806 1.00
## y_rep[211] 139.26 0.14 5.88 131.89 139.18 146.70 1732 1.00
## y_rep[212] 157.87 0.13 5.77 150.50 157.83 165.25 1945 1.00
## y_rep[213] 120.92 0.13 5.66 113.63 120.88 128.20 1957 1.00
## y_rep[214] 152.93 0.14 5.88 145.58 152.89 160.68 1867 1.00
## y_rep[215] 159.70 0.13 5.91 152.13 159.57 167.47 1977 1.00
## y_rep[216] 153.39 0.14 5.92 145.91 153.28 161.09 1856 1.00
## y_rep[217] 107.62 0.13 5.85 100.51 107.57 114.92 1928 1.00
## y_rep[218] 160.30 0.14 5.81 152.73 160.44 167.68 1789 1.00
## y_rep[219] 156.89 0.13 5.77 149.58 156.89 164.45 1989 1.00
## y_rep[220] 156.62 0.13 5.78 149.28 156.72 164.08 2008 1.00
## y_rep[221] 98.35 0.14 5.86 90.79 98.40 105.78 1874 1.00
## y_rep[222] 111.86 0.13 5.76 104.57 111.93 119.13 1922 1.00
## y_rep[223] 147.15 0.12 5.74 139.86 147.20 154.26 2154 1.00
## y_rep[224] 157.18 0.13 5.67 149.83 157.09 164.52 2037 1.00
## y_rep[225] 158.86 0.13 5.76 151.33 158.86 166.09 1944 1.00
## y_rep[226] 157.33 0.13 5.70 149.92 157.35 164.70 1962 1.00
## y_rep[227] 116.58 0.14 5.82 109.05 116.82 123.91 1637 1.00
## y_rep[228] 158.97 0.14 5.86 151.29 159.19 166.18 1764 1.00
## y_rep[229] 153.60 0.13 5.78 146.34 153.56 161.13 2028 1.00
## y_rep[230] 150.26 0.13 5.78 142.87 150.24 157.55 1958 1.00
## y_rep[231] 154.37 0.13 5.90 146.54 154.38 161.99 1911 1.00
## y_rep[232] 107.92 0.13 5.89 100.52 108.09 115.33 2042 1.00
## y_rep[233] 150.41 0.13 5.87 142.98 150.28 158.02 2205 1.00
## y_rep[234] 159.22 0.13 5.66 152.05 159.18 166.67 2001 1.00
## y_rep[235] 150.82 0.13 5.76 143.33 150.82 158.22 1861 1.00
## y_rep[236] 152.25 0.14 5.79 144.92 152.27 159.74 1752 1.00
## y_rep[237] 159.68 0.14 6.06 152.12 159.57 167.51 1876 1.00
## y_rep[238] 160.41 0.13 5.70 153.23 160.43 167.61 2073 1.00
## y_rep[239] 157.53 0.12 5.65 150.14 157.62 164.76 2112 1.00
## y_rep[240] 152.90 0.13 5.71 145.79 152.97 160.06 1882 1.00
## y_rep[241] 131.52 0.12 5.57 124.36 131.52 138.59 2060 1.00
## y_rep[242] 151.54 0.13 5.76 144.14 151.69 158.91 2104 1.00
## y_rep[243] 115.83 0.13 5.63 108.65 115.72 123.02 1825 1.00
## y_rep[244] 155.43 0.13 5.78 147.80 155.57 162.57 1995 1.00
## y_rep[245] 140.75 0.13 5.61 133.72 140.76 148.40 1850 1.00
## y_rep[246] 116.75 0.13 5.73 109.42 116.82 123.87 2100 1.00
## y_rep[247] 157.29 0.13 5.78 149.91 157.35 164.83 1925 1.00
## y_rep[248] 149.87 0.13 5.81 142.45 149.82 157.25 1947 1.00
## y_rep[249] 103.10 0.14 5.84 95.90 103.11 110.52 1863 1.00
## y_rep[250] 147.37 0.12 5.69 139.92 147.27 154.62 2083 1.00
## y_rep[251] 153.26 0.13 5.67 145.84 153.41 160.06 1868 1.00
## y_rep[252] 158.99 0.13 5.71 151.96 158.85 166.40 2039 1.00
## y_rep[253] 158.50 0.14 5.88 150.69 158.54 165.88 1727 1.00
## y_rep[254] 157.59 0.13 5.80 150.09 157.73 164.69 2036 1.00
## y_rep[255] 153.60 0.13 5.94 145.84 153.73 161.23 2115 1.00
## y_rep[256] 147.91 0.13 5.74 140.55 148.08 155.24 1937 1.00
## y_rep[257] 155.41 0.13 5.69 148.15 155.55 162.79 2002 1.00
## y_rep[258] 159.11 0.13 5.72 151.84 159.12 166.35 2069 1.00
## y_rep[259] 148.23 0.13 5.62 141.00 148.16 155.45 1867 1.00
## y_rep[260] 160.10 0.13 6.00 152.72 160.00 167.37 2032 1.00
## y_rep[261] 150.75 0.13 5.75 143.65 150.69 158.21 1827 1.00
## y_rep[262] 158.56 0.13 5.80 151.12 158.60 166.14 1849 1.00
## y_rep[263] 147.38 0.14 5.85 139.86 147.47 154.80 1787 1.00
## y_rep[264] 97.98 0.14 5.90 90.55 98.01 105.28 1789 1.00
## y_rep[265] 153.64 0.13 5.74 146.42 153.69 161.04 2000 1.00
## y_rep[266] 100.83 0.14 5.87 93.43 100.78 108.35 1750 1.00
## y_rep[267] 156.53 0.14 5.85 149.12 156.48 164.14 1771 1.00
## y_rep[268] 159.89 0.13 5.64 152.58 159.79 166.94 1753 1.00
## y_rep[269] 160.19 0.13 5.80 152.61 160.32 167.65 1932 1.00
## y_rep[270] 149.92 0.13 5.69 142.48 149.89 157.07 1986 1.00
## y_rep[271] 146.39 0.14 5.81 139.18 146.28 154.02 1848 1.00
## y_rep[272] 159.82 0.14 6.00 152.14 159.68 167.80 1865 1.00
## y_rep[273] 118.22 0.12 5.68 111.02 118.31 125.45 2114 1.00
## y_rep[274] 156.41 0.14 5.74 149.11 156.30 163.88 1593 1.00
## y_rep[275] 156.63 0.13 5.79 149.40 156.76 163.86 1935 1.00
## y_rep[276] 156.60 0.12 5.70 149.43 156.58 163.95 2163 1.00
## y_rep[277] 156.64 0.13 5.82 149.34 156.67 164.20 1963 1.00
## y_rep[278] 152.88 0.13 5.82 145.41 152.67 160.24 1955 1.00
## y_rep[279] 156.25 0.13 5.80 148.67 156.13 163.68 2001 1.00
## y_rep[280] 125.78 0.14 5.82 118.30 125.73 133.37 1854 1.00
## y_rep[281] 144.81 0.14 5.80 137.53 144.86 152.13 1824 1.00
## y_rep[282] 142.08 0.14 5.95 134.49 142.10 149.71 1929 1.00
## y_rep[283] 147.11 0.14 5.85 139.65 147.08 154.51 1858 1.00
## y_rep[284] 159.46 0.13 5.75 152.17 159.56 166.77 2014 1.00
## y_rep[285] 160.44 0.13 5.71 153.22 160.40 167.76 2001 1.00
## y_rep[286] 148.76 0.13 5.78 141.42 148.78 155.99 1969 1.00
## y_rep[287] 152.92 0.13 5.62 145.81 152.72 160.40 1921 1.00
## y_rep[288] 159.81 0.14 5.89 152.19 159.88 167.34 1755 1.00
## y_rep[289] 156.03 0.13 5.77 148.59 156.08 163.39 1981 1.00
## y_rep[290] 143.85 0.13 5.83 136.11 144.00 151.45 2048 1.00
## y_rep[291] 117.94 0.13 5.69 110.86 117.79 125.52 1962 1.00
## y_rep[292] 105.04 0.13 5.85 97.43 105.19 112.58 2180 1.00
## y_rep[293] 159.56 0.13 5.72 152.13 159.68 166.95 2005 1.00
## y_rep[294] 143.53 0.13 5.90 136.17 143.54 151.08 1994 1.00
## y_rep[295] 155.85 0.14 5.75 148.56 156.03 163.15 1808 1.00
## y_rep[296] 158.69 0.13 5.58 151.49 158.75 165.74 1943 1.00
## y_rep[297] 155.01 0.13 5.68 147.67 154.98 162.17 1824 1.00
## y_rep[298] 145.48 0.13 5.65 138.39 145.53 152.76 1966 1.00
## y_rep[299] 124.55 0.13 5.75 117.28 124.40 131.87 2006 1.00
## y_rep[300] 144.21 0.13 5.78 136.87 144.21 151.55 2014 1.00
## y_rep[301] 90.72 0.13 5.73 83.32 90.90 97.99 1845 1.00
## y_rep[302] 153.61 0.13 5.77 146.30 153.58 161.13 1854 1.00
## y_rep[303] 151.76 0.13 5.86 144.31 151.82 159.22 2045 1.00
## y_rep[304] 147.06 0.14 6.03 139.55 147.12 154.82 1924 1.00
## y_rep[305] 159.47 0.13 5.69 152.11 159.59 166.66 1990 1.00
## y_rep[306] 157.58 0.13 5.86 150.12 157.53 165.00 1910 1.00
## y_rep[307] 127.39 0.13 5.85 119.83 127.59 134.90 1962 1.00
## y_rep[308] 101.09 0.13 5.69 93.83 100.97 108.44 2050 1.00
## y_rep[309] 159.56 0.14 5.71 152.20 159.61 166.95 1628 1.00
## y_rep[310] 159.46 0.13 5.88 151.82 159.26 166.96 2015 1.00
## y_rep[311] 155.74 0.12 5.65 148.67 155.71 162.90 2169 1.00
## y_rep[312] 108.70 0.14 6.06 100.96 108.85 116.25 1985 1.00
## y_rep[313] 160.12 0.13 5.70 152.94 160.15 167.27 1936 1.00
## y_rep[314] 159.70 0.13 5.79 152.27 159.72 167.09 2047 1.00
## y_rep[315] 144.57 0.14 5.83 137.02 144.55 152.04 1724 1.00
## y_rep[316] 151.78 0.13 5.72 144.47 151.78 159.00 1997 1.00
## y_rep[317] 159.32 0.13 5.81 151.89 159.33 166.90 1982 1.00
## y_rep[318] 152.49 0.13 5.86 144.96 152.43 159.99 1988 1.00
## y_rep[319] 159.98 0.13 5.87 152.34 160.14 167.45 2036 1.00
## y_rep[320] 152.08 0.13 5.88 144.50 152.33 159.39 2096 1.00
## y_rep[321] 156.12 0.13 5.77 148.69 156.18 163.25 2071 1.00
## y_rep[322] 160.30 0.13 5.79 152.88 160.29 167.56 1892 1.00
## y_rep[323] 156.90 0.13 5.82 149.44 156.77 164.07 1988 1.00
## y_rep[324] 88.84 0.13 5.69 81.43 88.88 96.16 1979 1.00
## y_rep[325] 89.59 0.13 5.80 82.17 89.61 97.11 1922 1.00
## y_rep[326] 69.30 0.13 5.71 61.87 69.39 76.67 1928 1.00
## y_rep[327] 90.98 0.13 5.89 83.49 90.95 98.40 1908 1.00
## y_rep[328] 109.91 0.13 5.58 102.82 109.93 116.87 1952 1.00
## y_rep[329] 88.82 0.13 5.73 81.79 88.80 96.22 1994 1.00
## y_rep[330] 153.64 0.12 5.68 146.12 153.67 161.06 2079 1.00
## y_rep[331] 104.10 0.13 5.97 96.37 104.25 111.83 1974 1.00
## y_rep[332] 72.46 0.14 5.94 64.81 72.37 80.15 1888 1.00
## y_rep[333] 90.20 0.13 5.72 82.78 90.28 97.73 1883 1.00
## y_rep[334] 100.11 0.13 5.71 92.52 100.14 107.31 2010 1.00
## y_rep[335] 83.51 0.13 5.91 76.02 83.53 91.08 1950 1.00
## y_rep[336] 154.37 0.14 5.76 146.89 154.39 161.80 1816 1.00
## y_rep[337] 154.69 0.14 5.76 147.05 154.82 162.01 1709 1.00
## y_rep[338] 135.31 0.13 5.69 128.02 135.39 142.65 1913 1.00
## y_rep[339] 93.49 0.13 5.90 85.88 93.45 101.31 1988 1.00
## y_rep[340] 155.48 0.13 5.88 147.78 155.57 163.02 1999 1.00
## y_rep[341] 153.36 0.13 5.88 146.08 153.35 160.77 2161 1.00
## y_rep[342] 155.35 0.13 5.90 147.60 155.47 162.80 2083 1.00
## y_rep[343] 157.80 0.12 5.78 150.19 157.72 165.39 2245 1.00
## y_rep[344] 115.87 0.13 5.73 108.45 115.99 123.11 1825 1.00
## y_rep[345] 102.70 0.13 5.69 95.48 102.80 110.15 1863 1.00
## y_rep[346] 158.68 0.14 5.87 151.06 158.72 166.08 1836 1.00
## y_rep[347] 154.77 0.13 5.74 147.34 154.74 162.30 1840 1.00
## y_rep[348] 106.19 0.13 5.81 98.63 106.18 113.63 1943 1.00
## y_rep[349] 94.47 0.14 5.77 87.03 94.53 101.72 1600 1.00
## y_rep[350] 159.09 0.13 5.81 151.54 159.03 166.40 1903 1.00
## y_rep[351] 157.63 0.13 5.77 150.38 157.56 164.85 2011 1.00
## y_rep[352] 160.26 0.13 5.91 152.89 160.05 167.96 2090 1.00
## y_rep[353] 158.71 0.13 5.70 151.17 158.68 165.99 1891 1.00
## y_rep[354] 93.77 0.13 5.64 86.67 93.74 101.23 1884 1.00
## y_rep[355] 68.10 0.13 5.84 60.64 68.22 75.44 2007 1.00
## y_rep[356] 135.10 0.13 5.84 127.61 135.05 142.52 2079 1.00
## y_rep[357] 155.82 0.14 5.73 148.72 155.87 162.99 1712 1.00
## y_rep[358] 155.54 0.13 5.69 148.39 155.52 162.75 1930 1.00
## y_rep[359] 88.85 0.13 5.74 81.31 88.93 96.12 1913 1.00
## y_rep[360] 89.72 0.13 5.72 82.44 89.74 97.06 2051 1.00
## y_rep[361] 75.08 0.14 5.81 67.39 75.07 82.35 1782 1.00
## y_rep[362] 92.90 0.13 5.65 85.39 92.83 100.01 1908 1.00
## y_rep[363] 73.97 0.12 5.67 66.63 74.00 81.36 2107 1.00
## y_rep[364] 91.66 0.13 5.89 84.25 91.65 99.18 2049 1.00
## y_rep[365] 76.08 0.13 5.98 68.36 76.29 83.76 1991 1.00
## y_rep[366] 160.15 0.14 5.76 152.87 160.26 167.41 1745 1.00
## y_rep[367] 156.84 0.13 5.85 149.47 157.05 164.27 2018 1.00
## y_rep[368] 92.22 0.12 5.70 84.75 92.14 99.57 2137 1.00
## y_rep[369] 93.07 0.14 5.83 85.56 92.85 100.74 1758 1.00
## y_rep[370] 74.59 0.13 5.80 67.18 74.59 82.07 1944 1.00
## y_rep[371] 81.03 0.13 5.71 73.79 80.91 88.56 2017 1.00
## y_rep[372] 158.43 0.13 5.80 150.82 158.39 165.95 2023 1.00
## y_rep[373] 134.80 0.13 5.86 127.18 134.68 142.41 1928 1.00
## y_rep[374] 140.08 0.14 5.98 132.58 140.16 147.61 1773 1.00
## y_rep[375] 160.11 0.13 5.70 152.80 160.09 167.48 1963 1.00
## y_rep[376] 114.49 0.13 5.82 107.05 114.44 122.12 1970 1.00
## y_rep[377] 78.68 0.13 5.71 71.13 78.78 85.86 1952 1.00
## y_rep[378] 78.26 0.14 5.78 70.87 78.32 85.76 1780 1.00
## y_rep[379] 89.52 0.13 5.84 82.06 89.44 97.05 2079 1.00
## y_rep[380] 160.22 0.13 5.93 152.54 160.41 167.72 2115 1.00
## y_rep[381] 76.94 0.12 5.53 70.09 76.87 84.26 1958 1.00
## y_rep[382] 131.74 0.13 5.79 124.32 131.89 138.99 1971 1.00
## y_rep[383] 158.79 0.14 5.88 151.31 158.84 166.43 1875 1.00
## y_rep[384] 86.74 0.13 5.69 79.47 86.70 94.21 2013 1.00
## y_rep[385] 97.03 0.14 5.89 89.49 97.08 104.58 1777 1.00
## y_rep[386] 157.04 0.14 5.83 149.59 157.02 164.58 1731 1.00
## y_rep[387] 157.37 0.14 5.76 150.08 157.59 164.28 1643 1.00
## y_rep[388] 158.91 0.13 5.90 151.30 158.81 166.53 2158 1.00
## y_rep[389] 158.06 0.13 5.72 150.83 158.06 165.38 1856 1.00
## y_rep[390] 114.87 0.13 5.68 107.67 114.79 122.32 2034 1.00
## y_rep[391] 109.33 0.13 5.80 101.83 109.37 116.68 1913 1.00
## y_rep[392] 87.03 0.14 5.87 79.56 86.99 94.54 1783 1.00
## y_rep[393] 156.08 0.13 5.87 148.63 155.91 163.57 2086 1.00
## y_rep[394] 131.69 0.12 5.72 124.47 131.74 138.98 2217 1.00
## y_rep[395] 112.01 0.13 5.72 104.86 111.90 119.34 1827 1.00
## y_rep[396] 96.62 0.13 5.71 89.40 96.78 103.65 1836 1.00
## y_rep[397] 160.47 0.13 5.63 153.43 160.26 167.69 1935 1.00
## y_rep[398] 152.55 0.12 5.77 145.49 152.47 159.85 2158 1.00
## y_rep[399] 130.17 0.13 5.80 122.70 130.23 137.53 1987 1.00
## y_rep[400] 159.67 0.13 5.72 152.24 159.83 166.84 2002 1.00
## y_rep[401] 81.35 0.14 5.92 74.12 81.27 88.95 1811 1.00
## y_rep[402] 75.24 0.13 5.80 67.68 75.26 82.54 1881 1.00
## y_rep[403] 157.79 0.13 5.58 150.74 157.80 165.01 1788 1.00
## y_rep[404] 112.78 0.13 5.70 105.49 112.77 119.96 1953 1.00
## y_rep[405] 149.92 0.13 5.77 142.69 149.98 157.20 1895 1.00
## y_rep[406] 159.22 0.14 5.53 152.28 159.14 166.38 1626 1.00
## y_rep[407] 74.64 0.13 5.77 67.33 74.70 81.82 1887 1.00
## y_rep[408] 133.52 0.13 5.79 125.91 133.66 140.70 2076 1.00
## y_rep[409] 76.17 0.13 5.66 69.00 76.12 83.41 2002 1.00
## y_rep[410] 156.15 0.13 5.81 148.90 156.28 163.43 2115 1.00
## y_rep[411] 160.05 0.13 5.94 152.43 159.98 167.62 1980 1.00
## y_rep[412] 152.99 0.13 5.96 145.31 152.90 160.67 1961 1.00
## y_rep[413] 149.44 0.12 5.71 142.17 149.31 156.76 2168 1.00
## y_rep[414] 159.86 0.13 5.63 152.68 159.84 166.97 2001 1.00
## y_rep[415] 156.96 0.13 5.70 149.61 157.07 164.05 2027 1.00
## y_rep[416] 77.02 0.13 5.66 69.71 77.06 84.16 1974 1.00
## y_rep[417] 155.29 0.13 5.89 147.73 155.32 162.61 2022 1.00
## y_rep[418] 158.02 0.13 5.61 150.87 158.20 165.14 1965 1.00
## y_rep[419] 151.88 0.13 5.85 144.50 151.75 159.32 2169 1.00
## y_rep[420] 145.42 0.14 5.86 137.79 145.39 152.77 1850 1.00
## y_rep[421] 121.78 0.13 5.70 114.54 121.94 128.95 2078 1.00
## y_rep[422] 151.56 0.12 5.66 144.11 151.57 158.66 2320 1.00
## y_rep[423] 149.16 0.13 5.87 141.73 149.20 156.74 1995 1.00
## y_rep[424] 159.08 0.14 5.74 151.66 159.19 166.20 1614 1.00
## y_rep[425] 146.63 0.12 5.74 139.23 146.67 153.86 2193 1.00
## y_rep[426] 81.71 0.14 5.80 74.25 81.56 89.41 1823 1.00
## y_rep[427] 156.06 0.13 5.75 148.78 156.02 163.41 2012 1.00
## y_rep[428] 158.90 0.13 5.72 151.42 158.90 166.35 2018 1.00
## y_rep[429] 136.93 0.13 5.67 129.58 136.93 144.31 1949 1.00
## y_rep[430] 159.95 0.14 5.76 152.62 159.93 167.18 1768 1.00
## y_rep[431] 156.08 0.13 5.87 148.55 156.09 163.50 1951 1.00
## y_rep[432] 133.90 0.13 5.79 126.44 133.82 141.39 2004 1.00
## y_rep[433] 115.38 0.14 5.84 107.89 115.43 122.69 1789 1.00
## y_rep[434] 158.89 0.13 5.87 151.39 158.95 166.32 2037 1.00
## y_rep[435] 151.54 0.14 5.82 144.08 151.44 159.14 1712 1.00
## y_rep[436] 158.03 0.13 5.83 150.64 157.93 165.04 1873 1.00
## y_rep[437] 160.07 0.14 5.90 152.61 160.11 167.65 1876 1.00
## y_rep[438] 159.04 0.13 5.63 151.90 159.16 166.28 1954 1.00
## y_rep[439] 158.90 0.13 5.93 151.51 158.81 166.39 2196 1.00
## y_rep[440] 95.04 0.13 5.84 87.58 95.05 102.67 2135 1.00
## y_rep[441] 72.11 0.14 5.87 64.64 71.94 79.70 1878 1.00
## y_rep[442] 158.44 0.14 5.85 151.22 158.35 165.84 1845 1.00
## y_rep[443] 146.55 0.13 5.77 139.52 146.56 154.08 2008 1.00
## y_rep[444] 148.08 0.13 5.78 140.79 147.88 155.47 1998 1.00
## y_rep[445] 159.16 0.12 5.75 151.90 159.32 166.35 2147 1.00
## y_rep[446] 159.75 0.13 5.78 152.07 159.88 167.13 2069 1.00
## y_rep[447] 157.89 0.12 5.58 150.74 157.73 165.38 2026 1.00
## y_rep[448] 148.59 0.13 5.62 141.46 148.40 155.79 1784 1.00
## y_rep[449] 126.93 0.13 5.58 119.88 127.07 134.01 1824 1.00
## y_rep[450] 103.95 0.14 5.72 96.42 104.05 111.30 1710 1.00
## y_rep[451] 158.89 0.12 5.55 151.81 158.89 165.88 2062 1.00
## y_rep[452] 150.01 0.12 5.64 142.95 149.88 157.30 2133 1.00
## y_rep[453] 78.78 0.13 5.78 71.42 78.84 86.11 2020 1.00
## y_rep[454] 158.36 0.13 5.83 150.80 158.43 165.57 1984 1.00
## y_rep[455] 134.30 0.13 5.81 126.85 134.25 141.86 2002 1.00
## y_rep[456] 77.38 0.13 5.88 69.85 77.31 84.94 2030 1.00
## y_rep[457] 73.88 0.13 5.65 66.55 73.99 81.18 1843 1.00
## y_rep[458] 160.11 0.14 5.99 152.26 160.20 167.67 1953 1.00
## y_rep[459] 141.85 0.13 5.83 134.39 141.96 149.27 2148 1.00
## y_rep[460] 160.27 0.13 5.73 153.09 160.26 167.80 2003 1.00
## y_rep[461] 158.80 0.13 5.63 151.67 158.80 166.14 1972 1.00
## y_rep[462] 155.58 0.13 5.69 148.39 155.52 162.90 1941 1.00
## y_rep[463] 74.12 0.13 5.73 66.95 74.02 81.48 1909 1.00
## y_rep[464] 144.56 0.13 5.76 137.23 144.52 152.07 1977 1.00
## y_rep[465] 159.49 0.13 5.86 151.90 159.61 167.21 2007 1.00
## y_rep[466] 153.37 0.13 5.84 146.08 153.27 160.80 2017 1.00
## y_rep[467] 65.23 0.13 5.95 57.34 65.21 72.99 2060 1.00
## y_rep[468] 70.44 0.13 5.89 62.98 70.47 77.84 2111 1.00
## y_rep[469] 156.18 0.13 5.73 148.86 156.23 163.47 1989 1.00
## y_rep[470] 143.49 0.14 5.87 136.09 143.24 151.13 1761 1.00
## y_rep[471] 105.81 0.13 5.93 98.36 105.64 113.69 2063 1.00
## y_rep[472] 77.90 0.14 5.79 70.51 77.83 85.38 1650 1.00
## y_rep[473] 158.38 0.13 5.79 150.86 158.37 165.73 1973 1.00
## y_rep[474] 157.74 0.12 5.69 150.72 157.75 164.94 2098 1.00
## y_rep[475] 115.01 0.13 5.81 107.54 114.99 122.52 2124 1.00
## y_rep[476] 93.33 0.13 5.73 86.16 93.43 100.56 1999 1.00
## y_rep[477] 158.52 0.14 5.89 150.89 158.60 166.08 1863 1.00
## y_rep[478] 153.04 0.13 5.71 146.10 152.94 160.23 2025 1.00
## y_rep[479] 159.62 0.13 5.74 152.50 159.71 166.97 1935 1.00
## y_rep[480] 157.02 0.13 5.87 149.44 156.98 164.31 1951 1.00
## y_rep[481] 89.55 0.13 5.99 81.82 89.52 97.27 2018 1.00
## y_rep[482] 154.63 0.13 5.76 147.24 154.70 162.15 1904 1.00
## y_rep[483] 153.70 0.12 5.71 146.17 153.88 160.73 2188 1.00
## y_rep[484] 81.35 0.13 5.88 73.89 81.48 88.83 2016 1.00
## y_rep[485] 157.16 0.13 5.74 149.58 157.32 164.40 1927 1.00
## y_rep[486] 159.97 0.14 5.91 152.60 159.97 167.36 1876 1.00
## y_rep[487] 117.65 0.13 5.89 109.99 117.70 125.07 2153 1.00
## y_rep[488] 89.53 0.13 5.72 82.22 89.71 96.97 1932 1.00
## y_rep[489] 159.28 0.13 5.76 152.05 159.12 166.74 1834 1.00
## y_rep[490] 133.21 0.13 5.83 125.85 133.13 140.98 2074 1.00
## y_rep[491] 160.00 0.13 5.84 152.51 159.85 167.64 2012 1.00
## y_rep[492] 156.04 0.12 5.73 148.70 155.97 163.33 2260 1.00
## y_rep[493] 108.27 0.13 5.73 100.97 108.37 115.16 1801 1.00
## y_rep[494] 146.58 0.13 5.93 138.76 146.56 154.11 1937 1.00
## y_rep[495] 144.79 0.13 5.92 136.95 144.90 152.40 2032 1.00
## y_rep[496] 155.38 0.13 5.88 147.87 155.46 162.99 2003 1.00
## y_rep[497] 142.18 0.13 5.94 134.57 142.15 149.78 2098 1.00
## y_rep[498] 158.15 0.13 5.68 150.93 158.21 165.47 1898 1.00
## y_rep[499] 154.40 0.13 5.77 147.16 154.27 161.89 1931 1.00
## y_rep[500] 150.09 0.13 5.92 142.46 150.05 157.57 1925 1.00
## y_rep[501] 155.68 0.13 5.80 148.38 155.78 163.20 1887 1.00
## y_rep[502] 160.33 0.14 5.83 152.87 160.19 167.89 1773 1.00
## y_rep[503] 155.26 0.12 5.83 147.89 155.07 162.65 2252 1.00
## y_rep[504] 118.28 0.13 5.75 110.77 118.40 125.50 1934 1.00
## y_rep[505] 88.68 0.13 5.72 81.36 88.56 96.06 1919 1.00
## y_rep[506] 151.97 0.13 5.77 144.48 151.82 159.55 1873 1.00
## y_rep[507] 155.41 0.14 5.70 147.96 155.49 162.85 1725 1.00
## y_rep[508] 101.72 0.13 5.87 93.98 101.92 109.16 1956 1.00
## y_rep[509] 119.62 0.13 5.77 112.32 119.41 127.29 2013 1.00
## y_rep[510] 160.14 0.13 5.86 152.58 160.23 167.59 1912 1.00
## y_rep[511] 150.52 0.14 5.63 143.39 150.49 157.60 1697 1.00
## y_rep[512] 112.51 0.13 5.80 104.86 112.39 120.11 1905 1.00
## y_rep[513] 87.94 0.13 5.91 80.69 87.95 95.37 1931 1.00
## y_rep[514] 95.60 0.14 5.88 88.07 95.67 103.01 1894 1.00
## y_rep[515] 155.19 0.13 5.90 147.59 155.24 162.76 1968 1.00
## y_rep[516] 147.09 0.14 5.96 139.45 147.09 154.62 1938 1.00
## y_rep[517] 156.35 0.13 5.64 149.05 156.44 163.51 1959 1.00
## y_rep[518] 62.78 0.13 5.75 55.50 62.79 70.17 1949 1.00
## y_rep[519] 149.95 0.13 5.74 142.51 149.88 157.46 1839 1.00
## y_rep[520] 66.38 0.13 5.78 58.83 66.32 73.81 2072 1.00
## y_rep[521] 98.21 0.14 5.73 91.04 98.17 105.47 1776 1.00
## y_rep[522] 122.67 0.13 5.78 115.26 122.70 130.15 1869 1.00
## y_rep[523] 86.34 0.14 5.67 78.95 86.23 93.55 1597 1.00
## y_rep[524] 156.22 0.13 5.67 149.21 156.22 163.42 1960 1.00
## y_rep[525] 156.31 0.13 5.92 148.67 156.54 164.00 1956 1.00
## y_rep[526] 126.74 0.13 5.85 119.59 126.58 133.95 1896 1.00
## y_rep[527] 121.06 0.13 5.69 113.72 121.21 128.37 1998 1.00
## y_rep[528] 98.27 0.13 5.76 90.88 98.20 105.89 1958 1.00
## y_rep[529] 158.58 0.13 5.81 151.37 158.50 166.23 1959 1.00
## y_rep[530] 151.49 0.13 5.88 144.08 151.42 159.02 2117 1.00
## y_rep[531] 126.36 0.13 5.81 118.81 126.38 134.08 2118 1.00
## y_rep[532] 86.35 0.13 5.72 79.03 86.33 93.74 2086 1.00
## y_rep[533] 153.43 0.13 5.78 146.01 153.37 160.58 2019 1.00
## y_rep[534] 153.00 0.13 5.69 145.66 153.09 160.29 1909 1.00
## y_rep[535] 158.08 0.13 5.82 150.85 157.88 165.79 2125 1.00
## y_rep[536] 107.68 0.13 5.62 100.36 107.66 115.01 1827 1.00
## y_rep[537] 74.14 0.14 5.94 66.59 74.15 81.52 1911 1.00
## y_rep[538] 144.84 0.15 5.75 137.67 144.70 152.34 1543 1.00
## y_rep[539] 77.10 0.12 5.50 70.10 76.95 84.45 2129 1.00
## y_rep[540] 139.46 0.13 5.86 131.94 139.47 146.90 2057 1.00
## y_rep[541] 160.00 0.14 5.82 152.86 159.99 167.30 1740 1.00
## y_rep[542] 160.28 0.13 5.72 153.00 160.41 167.60 1830 1.00
## y_rep[543] 76.92 0.15 5.84 69.47 76.80 84.53 1470 1.00
## y_rep[544] 160.07 0.13 5.73 152.56 160.13 167.42 1995 1.00
## lp__ -1225.53 0.05 1.38 -1227.32 -1225.20 -1224.11 815 1.01
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:57:43 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).summary(lm(height~weight.z+I(weight.z^2),data=howell1))##
## Call:
## lm(formula = height ~ weight.z + I(weight.z^2), data = howell1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.9689 -3.9794 0.2364 3.9262 19.5182
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 146.6604 0.3748 391.30 <2e-16 ***
## weight.z 21.4149 0.2908 73.64 <2e-16 ***
## I(weight.z^2) -8.4123 0.2822 -29.80 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.766 on 541 degrees of freedom
## Multiple R-squared: 0.9565, Adjusted R-squared: 0.9564
## F-statistic: 5952 on 2 and 541 DF, p-value: < 2.2e-16Figure 4.9 of Statistical rethinking
6 One continious variable + one categorical variable
6.1 Howell 2: subset with only individuals over 18 years old
ggplot(howell2, aes(y = height, x = weight, group = male)) + geom_point() + geom_smooth(method = "lm")
Xmat <- model.matrix(~weight.z + male, howell2)
m8.data <- with(howell2, list(y = height, X = Xmat, nX = ncol(Xmat), n = nrow(howell2)))data {
int<lower=1> n;
int<lower=1> nX;
vector [n] y;
matrix [n,nX] X;
}
parameters {
vector[nX] beta;
real sigma;
}
transformed parameters {
vector[n] mu;
mu = X*beta;
}
model {
y~normal(mu,sigma);
beta ~ normal(0,100);
sigma~cauchy(0,5);
}
generated quantities {
vector[n] log_lik;
for (i in 1:n) {
log_lik[i] = normal_lpdf(y[i] | mu[i], sigma);
}
}fit.m8 <- sampling(m8, data = m8.data, iter = 1000, chains = 4, cores = 4)
print(fit.m8, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: ff983e7d557e1f552144a41483b33c4a.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 151.56 0.01 0.34 151.13 151.56 152.00 1318 1
## beta[2] 4.14 0.01 0.27 3.79 4.13 4.49 1313 1
## beta[3] 6.50 0.02 0.54 5.79 6.51 7.20 1188 1
## sigma 4.28 0.00 0.16 4.07 4.28 4.49 1917 1
## mu[1] 159.87 0.01 0.34 159.43 159.87 160.30 2005 1
## mu[2] 146.11 0.01 0.39 145.62 146.10 146.62 1832 1
## mu[3] 143.15 0.01 0.52 142.48 143.14 143.83 1729 1
## mu[4] 163.21 0.01 0.39 162.70 163.22 163.72 1971 1
## mu[5] 149.18 0.01 0.31 148.78 149.18 149.58 1612 1
## mu[6] 169.59 0.02 0.70 168.69 169.60 170.49 1482 1
## mu[7] 147.23 0.01 0.35 146.79 147.23 147.70 1812 1
## mu[8] 164.78 0.01 0.45 164.18 164.79 165.35 1717 1
## mu[9] 145.07 0.01 0.43 144.54 145.07 145.64 1808 1
## mu[10] 164.14 0.01 0.43 163.58 164.15 164.69 1809 1
## mu[11] 154.70 0.01 0.46 154.10 154.70 155.29 1190 1
## mu[12] 155.64 0.01 0.45 155.05 155.64 156.21 1400 1
## mu[13] 145.82 0.01 0.40 145.32 145.81 146.34 1828 1
## mu[14] 153.29 0.01 0.40 152.78 153.29 153.80 1220 1
## mu[15] 144.42 0.01 0.46 143.84 144.42 145.02 1783 1
## mu[16] 160.34 0.01 0.34 159.91 160.35 160.77 2082 1
## mu[17] 156.35 0.01 0.42 155.80 156.35 156.88 1455 1
## mu[18] 147.31 0.01 0.35 146.86 147.31 147.77 1808 1
## mu[19] 160.62 0.01 0.34 160.18 160.62 161.04 2119 1
## mu[20] 150.10 0.01 0.32 149.70 150.10 150.50 1481 1
## mu[21] 151.97 0.02 0.64 151.14 151.97 152.80 1279 1
## mu[22] 147.02 0.01 0.35 146.57 147.01 147.49 1821 1
## mu[23] 145.45 0.01 0.41 144.94 145.45 145.99 1819 1
## mu[24] 155.06 0.01 0.48 154.44 155.06 155.67 1366 1
## mu[25] 164.54 0.01 0.44 163.95 164.55 165.11 1749 1
## mu[26] 146.76 0.01 0.36 146.30 146.76 147.25 1828 1
## mu[27] 147.40 0.01 0.34 146.96 147.40 147.85 1803 1
## mu[28] 152.59 0.01 0.37 152.13 152.59 153.07 1249 1
## mu[29] 147.62 0.01 0.34 147.18 147.62 148.06 1789 1
## mu[30] 145.53 0.01 0.41 145.02 145.52 146.06 1821 1
## mu[31] 153.10 0.01 0.39 152.60 153.10 153.60 1227 1
## mu[32] 159.91 0.01 0.34 159.47 159.91 160.33 2011 1
## mu[33] 160.89 0.01 0.34 160.45 160.89 161.32 2150 1
## mu[34] 160.87 0.01 0.34 160.43 160.87 161.30 2148 1
## mu[35] 165.47 0.01 0.48 164.84 165.48 166.09 1647 1
## mu[36] 147.80 0.01 0.33 147.37 147.80 148.24 1775 1
## mu[37] 154.70 0.01 0.46 154.10 154.70 155.29 1190 1
## mu[38] 155.66 0.01 0.45 155.07 155.66 156.23 1401 1
## mu[39] 147.43 0.01 0.34 146.99 147.43 147.89 1801 1
## mu[40] 156.39 0.01 0.42 155.85 156.40 156.93 1460 1
## mu[41] 157.84 0.01 0.37 157.35 157.84 158.31 1640 1
## mu[42] 166.77 0.01 0.55 166.06 166.79 167.47 1577 1
## mu[43] 152.67 0.01 0.37 152.20 152.67 153.15 1246 1
## mu[44] 151.07 0.01 0.33 150.65 151.07 151.49 1365 1
## mu[45] 155.34 0.01 0.49 154.72 155.34 155.97 1185 1
## mu[46] 157.54 0.02 0.61 156.75 157.53 158.32 1174 1
## mu[47] 158.63 0.01 0.35 158.17 158.63 159.08 1781 1
## mu[48] 153.50 0.01 0.41 152.98 153.50 154.02 1214 1
## mu[49] 162.69 0.01 0.38 162.19 162.70 163.16 2072 1
## mu[50] 153.25 0.01 0.40 152.74 153.25 153.75 1222 1
## mu[51] 151.98 0.01 0.35 151.54 151.98 152.43 1286 1
## mu[52] 149.90 0.01 0.31 149.50 149.90 150.30 1508 1
## mu[53] 152.60 0.02 0.61 151.82 152.61 153.40 1289 1
## mu[54] 165.07 0.01 0.47 164.46 165.08 165.66 1679 1
## mu[55] 146.56 0.01 0.37 146.10 146.55 147.05 1831 1
## mu[56] 160.18 0.01 0.34 159.74 160.18 160.60 2057 1
## mu[57] 146.65 0.01 0.37 146.19 146.65 147.14 1830 1
## mu[58] 159.45 0.01 0.34 159.01 159.45 159.89 1930 1
## mu[59] 153.03 0.01 0.39 152.54 153.03 153.52 1230 1
## mu[60] 166.10 0.01 0.51 165.43 166.12 166.76 1610 1
## mu[61] 147.03 0.01 0.35 146.59 147.03 147.51 1820 1
## mu[62] 152.98 0.01 0.39 152.49 152.97 153.47 1232 1
## mu[63] 156.91 0.01 0.40 156.38 156.91 157.42 1512 1
## mu[64] 161.43 0.01 0.35 160.99 161.44 161.87 2189 1
## mu[65] 147.96 0.01 0.33 147.54 147.96 148.39 1760 1
## mu[66] 161.71 0.01 0.35 161.24 161.72 162.15 2197 1
## mu[67] 148.00 0.01 0.33 147.58 148.00 148.43 1757 1
## mu[68] 145.98 0.01 0.39 145.49 145.97 146.50 1830 1
## mu[69] 158.87 0.01 0.35 158.42 158.87 159.31 1823 1
## mu[70] 149.81 0.01 0.31 149.41 149.81 150.21 1521 1
## mu[71] 159.74 0.01 0.34 159.31 159.75 160.17 1983 1
## mu[72] 164.92 0.01 0.46 164.32 164.94 165.51 1698 1
## mu[73] 160.34 0.01 0.34 159.91 160.35 160.77 2082 1
## mu[74] 150.16 0.01 0.32 149.75 150.16 150.56 1474 1
## mu[75] 145.20 0.01 0.42 144.67 145.20 145.75 1812 1
## mu[76] 159.10 0.02 0.70 158.21 159.08 159.99 1177 1
## mu[77] 155.57 0.01 0.51 154.93 155.56 156.21 1183 1
## mu[78] 159.49 0.01 0.34 159.05 159.49 159.92 1936 1
## mu[79] 148.96 0.01 0.32 148.56 148.96 149.37 1643 1
## mu[80] 143.69 0.01 0.50 143.06 143.69 144.34 1752 1
## mu[81] 165.79 0.01 0.50 165.15 165.81 166.43 1627 1
## mu[82] 156.76 0.01 0.41 156.23 156.77 157.28 1496 1
## mu[83] 148.32 0.01 0.32 147.91 148.32 148.74 1723 1
## mu[84] 158.67 0.01 0.35 158.21 158.67 159.12 1787 1
## mu[85] 149.07 0.01 0.31 148.67 149.07 149.48 1627 1
## mu[86] 159.72 0.01 0.34 159.29 159.73 160.16 1979 1
## mu[87] 146.76 0.01 0.36 146.30 146.76 147.25 1828 1
## mu[88] 158.07 0.01 0.37 157.59 158.07 158.53 1680 1
## mu[89] 149.81 0.01 0.31 149.41 149.81 150.21 1521 1
## mu[90] 164.38 0.01 0.44 163.80 164.38 164.94 1773 1
## mu[91] 163.52 0.01 0.40 162.99 163.53 164.04 1914 1
## mu[92] 162.67 0.01 0.37 162.18 162.68 163.14 2075 1
## mu[93] 149.90 0.01 0.31 149.50 149.90 150.30 1508 1
## mu[94] 154.50 0.01 0.45 153.92 154.50 155.07 1193 1
## mu[95] 155.96 0.01 0.44 155.40 155.97 156.53 1424 1
## mu[96] 165.12 0.01 0.47 164.51 165.13 165.72 1673 1
## mu[97] 149.74 0.01 0.31 149.34 149.74 150.14 1532 1
## mu[98] 163.25 0.01 0.39 162.73 163.26 163.76 1964 1
## mu[99] 161.41 0.01 0.35 160.97 161.42 161.86 2188 1
## mu[100] 149.98 0.01 0.31 149.57 149.97 150.37 1498 1
## mu[101] 160.69 0.01 0.34 160.25 160.69 161.12 2128 1
## mu[102] 147.40 0.01 0.34 146.96 147.40 147.85 1803 1
## mu[103] 154.65 0.01 0.46 154.05 154.64 155.23 1191 1
## mu[104] 167.52 0.01 0.59 166.76 167.53 168.27 1545 1
## mu[105] 159.94 0.01 0.34 159.51 159.95 160.37 2018 1
## mu[106] 147.91 0.01 0.33 147.48 147.91 148.34 1765 1
## mu[107] 146.36 0.01 0.38 145.89 146.35 146.86 1832 1
## mu[108] 149.72 0.01 0.31 149.32 149.72 150.12 1534 1
## mu[109] 157.78 0.01 0.37 157.29 157.78 158.26 1631 1
## mu[110] 153.39 0.01 0.40 152.88 153.39 153.90 1217 1
## mu[111] 164.59 0.01 0.44 164.01 164.60 165.16 1741 1
## mu[112] 143.87 0.01 0.49 143.25 143.87 144.50 1760 1
## mu[113] 148.92 0.01 0.32 148.52 148.93 149.33 1647 1
## mu[114] 162.03 0.01 0.36 161.56 162.04 162.49 2194 1
## mu[115] 160.65 0.01 0.34 160.21 160.66 161.08 2124 1
## mu[116] 156.97 0.02 0.58 156.22 156.96 157.71 1175 1
## mu[117] 163.89 0.01 0.42 163.33 163.89 164.42 1851 1
## mu[118] 164.70 0.01 0.45 164.11 164.71 165.28 1726 1
## mu[119] 149.80 0.01 0.31 149.39 149.79 150.19 1524 1
## mu[120] 148.96 0.01 0.32 148.56 148.96 149.37 1643 1
## mu[121] 161.07 0.01 0.34 160.63 161.08 161.50 2167 1
## mu[122] 151.14 0.01 0.33 150.72 151.14 151.56 1357 1
## mu[123] 158.67 0.01 0.35 158.21 158.67 159.12 1787 1
## mu[124] 149.61 0.01 0.31 149.21 149.61 150.01 1550 1
## mu[125] 155.72 0.01 0.51 155.07 155.71 156.37 1181 1
## mu[126] 151.00 0.01 0.33 150.58 150.99 151.41 1372 1
## mu[127] 143.36 0.01 0.51 142.71 143.36 144.03 1738 1
## mu[128] 165.59 0.01 0.49 164.96 165.61 166.22 1639 1
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## log_lik[296] -2.47 0.00 0.05 -2.53 -2.46 -2.40 1952 1
## log_lik[297] -2.38 0.00 0.04 -2.43 -2.38 -2.34 1885 1
## log_lik[298] -3.64 0.00 0.17 -3.85 -3.63 -3.42 1342 1
## log_lik[299] -5.65 0.01 0.30 -6.04 -5.64 -5.27 2364 1
## log_lik[300] -2.40 0.00 0.04 -2.45 -2.40 -2.35 2030 1
## log_lik[301] -2.40 0.00 0.04 -2.45 -2.40 -2.35 1916 1
## log_lik[302] -3.23 0.00 0.11 -3.37 -3.23 -3.10 2276 1
## log_lik[303] -2.58 0.00 0.06 -2.66 -2.58 -2.51 1951 1
## log_lik[304] -2.53 0.00 0.05 -2.60 -2.53 -2.47 1798 1
## log_lik[305] -2.40 0.00 0.04 -2.46 -2.40 -2.35 1982 1
## log_lik[306] -2.80 0.00 0.09 -2.92 -2.80 -2.69 1860 1
## log_lik[307] -3.16 0.00 0.10 -3.29 -3.16 -3.03 2013 1
## log_lik[308] -2.49 0.00 0.04 -2.55 -2.49 -2.44 1708 1
## log_lik[309] -2.43 0.00 0.05 -2.50 -2.43 -2.37 1908 1
## log_lik[310] -2.58 0.00 0.07 -2.68 -2.58 -2.49 1140 1
## log_lik[311] -2.39 0.00 0.04 -2.43 -2.39 -2.34 1898 1
## log_lik[312] -2.39 0.00 0.04 -2.44 -2.39 -2.34 1834 1
## log_lik[313] -2.41 0.00 0.04 -2.46 -2.41 -2.36 1785 1
## log_lik[314] -2.41 0.00 0.04 -2.47 -2.41 -2.36 1448 1
## log_lik[315] -3.34 0.00 0.12 -3.50 -3.34 -3.19 1881 1
## log_lik[316] -2.44 0.00 0.04 -2.50 -2.44 -2.39 1615 1
## log_lik[317] -15.38 0.02 1.03 -16.76 -15.37 -14.10 1956 1
## log_lik[318] -3.31 0.00 0.12 -3.46 -3.30 -3.16 1311 1
## log_lik[319] -2.43 0.00 0.05 -2.49 -2.43 -2.37 1889 1
## log_lik[320] -3.33 0.00 0.10 -3.46 -3.33 -3.19 1678 1
## log_lik[321] -2.38 0.00 0.04 -2.42 -2.38 -2.33 1918 1
## log_lik[322] -4.24 0.01 0.24 -4.56 -4.24 -3.94 1373 1
## log_lik[323] -2.39 0.00 0.04 -2.44 -2.39 -2.34 2019 1
## log_lik[324] -2.63 0.00 0.06 -2.72 -2.63 -2.55 2104 1
## log_lik[325] -2.84 0.00 0.07 -2.93 -2.84 -2.75 1429 1
## log_lik[326] -4.21 0.00 0.21 -4.49 -4.21 -3.94 1836 1
## log_lik[327] -4.40 0.00 0.18 -4.64 -4.40 -4.18 1713 1
## log_lik[328] -2.96 0.00 0.09 -3.08 -2.96 -2.85 1971 1
## log_lik[329] -3.13 0.00 0.12 -3.29 -3.12 -2.97 1510 1
## log_lik[330] -2.42 0.00 0.05 -2.49 -2.42 -2.36 1330 1
## log_lik[331] -2.89 0.00 0.09 -3.02 -2.89 -2.77 1548 1
## log_lik[332] -3.57 0.00 0.16 -3.77 -3.56 -3.37 2014 1
## log_lik[333] -2.38 0.00 0.04 -2.43 -2.38 -2.33 1835 1
## log_lik[334] -3.35 0.00 0.16 -3.56 -3.34 -3.14 1448 1
## log_lik[335] -2.91 0.00 0.08 -3.02 -2.91 -2.82 1449 1
## log_lik[336] -2.69 0.00 0.07 -2.79 -2.69 -2.60 2039 1
## log_lik[337] -2.40 0.00 0.04 -2.45 -2.40 -2.35 1928 1
## log_lik[338] -2.45 0.00 0.04 -2.50 -2.44 -2.39 1802 1
## log_lik[339] -2.60 0.00 0.07 -2.69 -2.60 -2.52 1961 1
## log_lik[340] -2.56 0.00 0.05 -2.63 -2.56 -2.50 1551 1
## log_lik[341] -2.38 0.00 0.04 -2.43 -2.38 -2.33 1831 1
## log_lik[342] -2.55 0.00 0.06 -2.63 -2.55 -2.48 1834 1
## log_lik[343] -3.14 0.00 0.10 -3.26 -3.13 -3.02 1374 1
## log_lik[344] -2.39 0.00 0.04 -2.44 -2.39 -2.34 1840 1
## log_lik[345] -2.48 0.00 0.05 -2.54 -2.48 -2.42 1639 1
## log_lik[346] -2.39 0.00 0.04 -2.44 -2.39 -2.34 1570 1
## log_lik[347] -2.71 0.00 0.06 -2.79 -2.71 -2.64 1742 1
## log_lik[348] -4.95 0.01 0.26 -5.29 -4.95 -4.62 1343 1
## log_lik[349] -2.47 0.00 0.05 -2.54 -2.46 -2.40 1666 1
## log_lik[350] -2.38 0.00 0.04 -2.42 -2.38 -2.33 1879 1
## log_lik[351] -2.42 0.00 0.05 -2.49 -2.41 -2.35 1113 1
## log_lik[352] -2.85 0.00 0.09 -2.96 -2.84 -2.74 2055 1
## lp__ -689.37 0.05 1.42 -691.22 -689.07 -687.86 818 1
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:58:53 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).summary(lm(height~weight.z+male,data=howell2))##
## Call:
## lm(formula = height ~ weight.z + male, data = howell2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.7859 -2.5506 0.4669 2.6278 14.1486
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 151.5501 0.3391 446.98 <2e-16 ***
## weight.z 4.1399 0.2678 15.46 <2e-16 ***
## male 6.5003 0.5359 12.13 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.272 on 349 degrees of freedom
## Multiple R-squared: 0.6973, Adjusted R-squared: 0.6955
## F-statistic: 401.9 on 2 and 349 DF, p-value: < 2.2e-166.2 Howell 1: data set with all individuals
howell1$male <- ifelse(howell1$male==1,2,1)
m8.1.data <- list(N = nrow(howell1), height = howell1$height, weight = howell1$weight.z, male = as.integer(howell1$male), n = length(unique(howell1$male)))data { // observed variables
int<lower=1> N; // number of observations
vector[N] height; // response variables
vector[N] weight; // predictor (previously centered)
int<lower=0> n;
int<lower=1,upper=n> male[N];
}
parameters { // unobserved variables
real beta[2]; // parameters: intercept and slope
vector[n] alpha;
real sigma;
}
transformed parameters {
vector[N] mu; // create one variable with predictions for each obs
for (i in 1:N) // loop for all cases
mu[i] = alpha[male[i]] + beta[1]*weight[i] + beta[2]*(weight[i] .* weight[i]);
}
model {
height ~ normal(mu,sigma); // likelihood
alpha ~ normal(178, 100); // prior of the intercept
beta[1] ~ normal(0, 100); // prior of the slope
beta[2] ~ normal(0, 100);
sigma ~ uniform(0,50);
}
fit.m8.1 <- sampling(m8.1, data = m8.1.data, iter = 1000, chains = 4, cores = 4)## Warning: There were 275 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## http://mc-stan.org/misc/warnings.html#bfmi-low## Warning: Examine the pairs() plot to diagnose sampling problemsprint(fit.m8.1, probs = c(0.10, 0.5, 0.9))## Inference for Stan model: 7f6f4063a593c9d5515aafcd1d091662.
## 4 chains, each with iter=1000; warmup=500; thin=1;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 10% 50% 90% n_eff Rhat
## beta[1] 20.80 0.07 1.04 20.16 20.73 21.32 222 1.03
## beta[2] -8.91 0.12 1.56 -9.56 -9.06 -8.31 179 1.04
## alpha[1] 144.86 0.27 3.24 143.62 145.26 145.89 142 1.04
## alpha[2] 149.38 0.18 2.32 148.37 149.56 150.49 166 1.04
## sigma 9.95 5.57 8.49 5.22 5.48 21.35 2 3.03
## mu[1] 160.50 0.05 0.93 159.77 160.53 161.23 366 1.00
## mu[2] 146.06 0.27 3.19 144.87 146.47 147.08 140 1.04
## mu[3] 138.99 0.28 3.35 137.77 139.39 140.05 145 1.04
## mu[4] 161.52 0.08 1.40 160.52 161.41 162.48 325 1.02
## mu[5] 151.54 0.28 2.74 150.43 151.89 152.47 99 1.04
## mu[6] 157.25 0.36 4.98 154.89 156.77 159.34 195 1.04
## mu[7] 148.29 0.26 3.06 147.16 148.67 149.29 137 1.04
## mu[8] 161.23 0.13 2.09 159.98 161.03 162.50 272 1.03
## mu[9] 143.79 0.27 3.28 142.55 144.20 144.83 143 1.04
## mu[10] 161.40 0.10 1.79 160.24 161.24 162.49 291 1.03
## mu[11] 156.66 0.14 1.53 155.59 156.84 157.58 121 1.04
## mu[12] 156.01 0.14 1.80 155.05 156.15 156.93 176 1.03
## mu[13] 145.45 0.27 3.22 144.24 145.85 146.47 141 1.04
## mu[14] 155.93 0.18 1.82 154.99 156.16 156.81 105 1.04
## mu[15] 142.24 0.28 3.32 141.02 142.65 143.29 144 1.04
## mu[16] 160.78 0.04 0.88 160.04 160.79 161.53 412 1.00
## mu[17] 157.01 0.12 1.66 156.11 157.13 157.90 183 1.03
## mu[18] 148.43 0.26 3.05 147.30 148.81 149.42 137 1.04
## mu[19] 116.26 0.13 1.69 115.29 116.36 117.32 173 1.03
## mu[20] 94.95 0.09 1.61 93.99 95.08 95.95 328 1.02
## mu[21] 83.41 0.05 1.40 82.34 83.46 84.61 755 1.00
## mu[22] 160.92 0.04 0.87 160.16 160.93 161.67 436 1.00
## mu[23] 152.83 0.29 2.56 151.78 153.15 153.73 78 1.04
## mu[24] 126.44 0.16 2.10 125.47 126.60 127.53 166 1.03
## mu[25] 101.30 0.12 1.91 100.20 101.50 102.25 267 1.03
## mu[26] 149.21 0.18 2.33 148.20 149.39 150.33 166 1.04
## mu[27] 147.88 0.26 3.09 146.80 148.26 148.88 138 1.04
## mu[28] 144.65 0.27 3.25 143.40 145.06 145.69 142 1.04
## mu[29] 134.85 0.18 2.36 133.88 135.02 135.99 165 1.04
## mu[30] 119.71 0.22 2.85 118.51 120.04 120.72 161 1.04
## mu[31] 111.34 0.11 1.50 110.34 111.39 112.36 180 1.02
## mu[32] 155.11 0.15 1.91 154.11 155.26 156.06 173 1.03
## mu[33] 161.30 0.12 1.97 160.08 161.12 162.47 281 1.03
## mu[34] 147.39 0.27 3.12 146.28 147.78 148.39 139 1.04
## mu[35] 148.60 0.26 3.04 147.47 148.97 149.59 136 1.04
## mu[36] 155.43 0.21 1.99 154.48 155.67 156.30 92 1.04
## mu[37] 148.99 0.26 3.01 147.87 149.36 149.97 135 1.04
## mu[38] 144.81 0.27 3.24 143.57 145.22 145.85 142 1.04
## mu[39] 155.80 0.18 1.86 154.88 156.03 156.68 103 1.04
## mu[40] 160.53 0.05 0.92 159.79 160.55 161.25 369 1.00
## mu[41] 161.05 0.04 0.87 160.26 161.05 161.80 452 1.00
## mu[42] 161.04 0.04 0.87 160.26 161.04 161.80 452 1.00
## mu[43] 160.94 0.16 2.44 159.58 160.71 162.31 240 1.03
## mu[44] 149.31 0.26 2.98 148.21 149.68 150.29 135 1.04
## mu[45] 156.66 0.14 1.53 155.59 156.84 157.58 121 1.04
## mu[46] 133.33 0.27 3.30 132.11 133.72 134.41 147 1.04
## mu[47] 118.40 0.14 1.78 117.41 118.52 119.49 171 1.03
## mu[48] 121.38 0.23 2.92 120.18 121.71 122.39 158 1.04
## mu[49] 92.76 0.08 1.53 91.78 92.87 93.80 367 1.02
## mu[50] 156.04 0.14 1.80 155.08 156.17 156.95 176 1.03
## mu[51] 148.66 0.26 3.03 147.53 149.04 149.65 136 1.04
## mu[52] 157.08 0.12 1.65 156.18 157.20 157.96 183 1.03
## mu[53] 158.79 0.09 1.33 157.98 158.87 159.61 220 1.02
## mu[54] 122.55 0.15 1.95 121.55 122.70 123.63 168 1.03
## mu[55] 99.57 0.11 1.82 98.53 99.75 100.53 280 1.03
## mu[56] 91.09 0.07 1.48 90.10 91.19 92.14 408 1.01
## mu[57] 160.14 0.22 3.17 158.49 159.83 161.66 216 1.04
## mu[58] 155.48 0.20 1.97 154.53 155.72 156.36 94 1.04
## mu[59] 153.98 0.29 2.35 152.97 154.27 154.86 65 1.04
## mu[60] 156.85 0.12 1.44 155.70 157.01 157.83 156 1.04
## mu[61] 156.91 0.12 1.56 155.46 156.96 158.17 157 1.04
## mu[62] 159.56 0.07 1.15 158.84 159.62 160.32 278 1.02
## mu[63] 156.06 0.17 1.77 155.15 156.28 156.95 106 1.04
## mu[64] 161.51 0.06 1.21 160.60 161.43 162.43 351 1.02
## mu[65] 155.90 0.18 1.83 154.96 156.13 156.77 104 1.04
## mu[66] 154.90 0.25 2.13 153.89 155.17 155.76 75 1.04
## mu[67] 152.57 0.29 2.60 151.55 152.89 153.47 82 1.04
## mu[68] 150.58 0.18 2.26 149.56 150.76 151.66 166 1.03
## mu[69] 161.12 0.14 2.23 159.82 160.91 162.43 256 1.03
## mu[70] 146.99 0.27 3.14 145.86 147.39 147.99 139 1.04
## mu[71] 128.72 0.17 2.18 127.74 128.89 129.82 166 1.03
## mu[72] 126.65 0.25 3.12 125.40 127.02 127.71 150 1.04
## mu[73] 160.69 0.04 0.89 159.95 160.71 161.43 396 1.00
## mu[74] 147.17 0.27 3.13 146.05 147.57 148.17 139 1.04
## mu[75] 134.87 0.27 3.33 133.66 135.26 135.95 147 1.04
## mu[76] 160.22 0.05 0.99 159.50 160.26 160.93 331 1.01
## mu[77] 155.75 0.19 1.88 154.81 155.98 156.63 102 1.04
## mu[78] 96.17 0.09 1.33 95.09 96.11 97.18 226 1.02
## mu[79] 160.59 0.19 2.79 159.10 160.33 162.04 226 1.04
## mu[80] 147.92 0.26 3.09 146.83 148.30 148.91 138 1.04
## mu[81] 152.12 0.28 2.67 151.06 152.45 153.03 89 1.04
## mu[82] 130.87 0.27 3.25 129.58 131.24 131.94 148 1.04
## mu[83] 155.71 0.19 1.89 154.77 155.95 156.59 102 1.04
## mu[84] 157.74 0.11 1.54 156.89 157.84 158.59 191 1.03
## mu[85] 161.26 0.04 0.92 160.47 161.23 162.03 446 1.00
## mu[86] 149.60 0.26 2.95 148.49 149.96 150.56 134 1.04
## mu[87] 110.19 0.11 1.46 109.17 110.24 111.20 183 1.02
## mu[88] 161.34 0.05 0.96 160.53 161.30 162.13 428 1.01
## mu[89] 149.66 0.26 2.95 148.55 150.02 150.62 134 1.04
## mu[90] 145.79 0.27 3.20 144.60 146.20 146.82 141 1.04
## mu[91] 159.77 0.06 1.10 159.04 159.82 160.51 292 1.01
## mu[92] 152.44 0.29 2.62 151.43 152.77 153.35 83 1.04
## mu[93] 160.42 0.05 0.95 159.69 160.46 161.14 354 1.01
## mu[94] 161.17 0.13 2.16 159.90 160.96 162.48 264 1.03
## mu[95] 109.55 0.16 2.34 108.38 109.81 110.50 204 1.03
## mu[96] 160.78 0.04 0.88 160.04 160.79 161.53 412 1.00
## mu[97] 152.90 0.29 2.55 151.85 153.22 153.80 77 1.04
## mu[98] 144.08 0.27 3.27 142.82 144.49 145.11 143 1.04
## mu[99] 111.42 0.11 1.50 110.42 111.47 112.44 180 1.03
## mu[100] 156.36 0.16 2.08 154.59 156.33 157.82 160 1.04
## mu[101] 135.09 0.18 2.36 134.12 135.26 136.23 164 1.04
## mu[102] 156.90 0.11 1.42 155.76 157.05 157.89 157 1.04
## mu[103] 160.25 0.05 0.99 159.52 160.29 160.96 334 1.01
## mu[104] 151.22 0.27 2.79 150.10 151.57 152.15 105 1.04
## mu[105] 140.42 0.28 3.34 139.19 140.83 141.48 145 1.04
## mu[106] 160.77 0.17 2.62 159.34 160.53 162.17 232 1.04
## mu[107] 157.56 0.11 1.57 156.70 157.67 158.42 189 1.03
## mu[108] 150.21 0.26 2.90 149.06 150.57 151.16 124 1.04
## mu[109] 140.00 0.28 3.35 138.77 140.40 141.06 145 1.04
## mu[110] 109.20 0.11 1.43 108.15 109.24 110.21 185 1.02
## mu[111] 107.35 0.10 1.38 106.30 107.39 108.37 190 1.02
## mu[112] 159.59 0.07 1.15 158.87 159.65 160.35 280 1.02
## mu[113] 151.38 0.27 2.77 150.28 151.73 152.31 102 1.04
## mu[114] 107.95 0.15 2.26 106.82 108.19 108.90 219 1.03
## mu[115] 103.04 0.09 1.29 102.01 103.05 104.06 206 1.02
## mu[116] 91.34 0.09 1.49 90.22 91.24 92.40 249 1.02
## mu[117] 160.41 0.05 0.95 159.68 160.44 161.12 353 1.01
## mu[118] 147.39 0.27 3.12 146.28 147.78 148.39 139 1.04
## mu[119] 159.03 0.08 1.28 158.25 159.11 159.83 234 1.02
## mu[120] 152.44 0.29 2.62 151.43 152.77 153.35 83 1.04
## mu[121] 161.35 0.11 1.90 160.16 161.17 162.50 285 1.03
## mu[122] 161.50 0.09 1.52 160.45 161.37 162.50 311 1.03
## mu[123] 161.51 0.06 1.21 160.60 161.43 162.42 352 1.02
## mu[124] 85.35 0.12 1.81 84.11 85.17 86.60 210 1.02
## mu[125] 152.57 0.29 2.60 151.55 152.89 153.47 82 1.04
## mu[126] 156.58 0.15 1.56 155.55 156.76 157.50 115 1.04
## mu[127] 96.73 0.09 1.32 95.66 96.67 97.74 226 1.02
## mu[128] 156.48 0.13 1.74 155.55 156.62 157.38 179 1.03
## mu[129] 161.09 0.14 2.26 159.79 160.88 162.42 253 1.03
## mu[130] 152.35 0.29 2.63 151.33 152.68 153.25 85 1.04
## mu[131] 111.12 0.17 2.42 109.94 111.39 112.07 193 1.03
## mu[132] 103.04 0.09 1.29 102.01 103.05 104.06 206 1.02
## mu[133] 161.52 0.08 1.41 160.51 161.41 162.48 323 1.02
## mu[134] 161.25 0.04 0.91 160.46 161.23 162.03 447 1.00
## mu[135] 152.66 0.29 2.59 151.64 152.99 153.56 80 1.04
## mu[136] 160.96 0.04 0.87 160.19 160.96 161.71 441 1.00
## mu[137] 148.60 0.26 3.04 147.47 148.97 149.59 136 1.04
## mu[138] 156.63 0.14 1.54 155.57 156.82 157.56 119 1.04
## mu[139] 138.85 0.19 2.43 137.86 139.04 140.00 164 1.04
## mu[140] 159.53 0.25 3.62 157.70 159.17 161.18 208 1.04
## mu[141] 87.59 0.06 1.41 86.58 87.67 88.70 538 1.01
## mu[142] 160.55 0.05 0.92 159.82 160.58 161.27 372 1.00
## mu[143] 149.50 0.26 2.96 148.41 149.87 150.47 134 1.04
## mu[144] 152.67 0.16 2.13 151.67 152.84 153.71 168 1.03
## mu[145] 118.01 0.22 2.77 116.79 118.33 119.00 164 1.04
## mu[146] 146.59 0.27 3.16 145.42 146.99 147.59 140 1.04
## mu[147] 152.32 0.29 2.64 151.30 152.65 153.23 85 1.04
## mu[148] 112.21 0.18 2.48 111.00 112.48 113.18 186 1.04
## mu[149] 108.20 0.10 1.40 107.14 108.23 109.21 188 1.02
## mu[150] 158.73 0.09 1.34 157.92 158.81 159.55 218 1.02
## mu[151] 156.00 0.18 1.80 155.08 156.22 156.88 105 1.04
## mu[152] 115.90 0.20 2.67 114.70 116.20 116.90 170 1.04
## mu[153] 114.28 0.12 1.61 113.32 114.36 115.32 175 1.03
## mu[154] 161.28 0.12 2.00 160.05 161.10 162.47 280 1.03
## mu[155] 140.89 0.28 3.34 139.65 141.29 141.94 145 1.04
## mu[156] 151.16 0.27 2.79 150.04 151.51 152.10 106 1.04
## mu[157] 92.76 0.08 1.53 91.78 92.87 93.80 367 1.02
## mu[158] 161.42 0.05 1.03 160.57 161.36 162.24 402 1.01
## mu[159] 160.94 0.04 0.87 160.17 160.95 161.69 439 1.00
## mu[160] 156.99 0.12 1.46 155.63 157.07 158.15 158 1.04
## mu[161] 161.45 0.10 1.68 160.35 161.31 162.52 298 1.03
## mu[162] 161.25 0.12 2.05 160.00 161.06 162.49 277 1.03
## mu[163] 152.42 0.29 2.62 151.41 152.75 153.32 84 1.04
## mu[164] 118.93 0.22 2.81 117.72 119.26 119.94 162 1.04
## mu[165] 151.22 0.27 2.79 150.10 151.57 152.15 105 1.04
## mu[166] 161.12 0.04 0.88 160.35 161.11 161.90 456 1.00
## mu[167] 154.06 0.29 2.33 153.06 154.34 154.93 65 1.04
## mu[168] 159.59 0.07 1.15 158.87 159.65 160.35 280 1.02
## mu[169] 152.17 0.28 2.66 151.13 152.50 153.08 88 1.04
## mu[170] 156.93 0.11 1.41 155.77 157.07 157.94 157 1.04
## mu[171] 91.09 0.07 1.48 90.10 91.19 92.14 408 1.01
## mu[172] 91.93 0.08 1.51 90.95 92.03 92.97 386 1.01
## mu[173] 153.90 0.29 2.36 152.88 154.19 154.78 65 1.04
## mu[174] 139.57 0.28 3.35 138.35 139.97 140.63 145 1.04
## mu[175] 160.88 0.16 2.51 159.49 160.64 162.26 237 1.03
## mu[176] 157.00 0.11 1.41 155.75 157.11 158.12 159 1.04
## mu[177] 146.06 0.27 3.19 144.87 146.47 147.08 140 1.04
## mu[178] 160.88 0.04 0.87 160.12 160.89 161.63 428 1.00
## mu[179] 160.81 0.17 2.58 159.40 160.57 162.20 234 1.04
## mu[180] 152.30 0.29 2.64 151.27 152.63 153.20 86 1.04
## mu[181] 161.24 0.04 0.91 160.46 161.22 162.02 449 1.00
## mu[182] 155.40 0.21 1.99 154.46 155.64 156.27 91 1.04
## mu[183] 158.19 0.32 4.46 156.05 157.74 160.11 199 1.04
## mu[184] 160.92 0.04 0.87 160.16 160.93 161.67 436 1.00
## mu[185] 142.91 0.19 2.45 141.95 143.12 144.07 165 1.04
## mu[186] 139.57 0.28 3.35 138.35 139.97 140.63 145 1.04
## mu[187] 122.33 0.24 2.96 121.12 122.67 123.36 156 1.04
## mu[188] 128.39 0.17 2.17 127.41 128.56 129.48 166 1.03
## mu[189] 121.29 0.15 1.90 120.28 121.43 122.38 169 1.03
## mu[190] 144.41 0.27 3.26 143.15 144.82 145.44 142 1.04
## mu[191] 159.42 0.07 1.19 158.69 159.48 160.18 270 1.02
## mu[192] 153.74 0.29 2.40 152.72 154.03 154.62 67 1.04
## mu[193] 154.85 0.25 2.15 153.84 155.12 155.71 74 1.04
## mu[194] 149.18 0.26 2.99 148.08 149.55 150.16 135 1.04
## mu[195] 151.44 0.27 2.76 150.33 151.78 152.37 101 1.04
## mu[196] 92.30 0.08 1.52 91.33 92.41 93.34 377 1.01
## mu[197] 154.66 0.26 2.19 153.70 154.93 155.53 71 1.04
## mu[198] 161.49 0.09 1.55 160.43 161.36 162.51 308 1.03
## mu[199] 156.99 0.11 1.40 155.75 157.11 158.06 158 1.04
## mu[200] 112.71 0.12 1.55 111.73 112.78 113.76 178 1.03
## mu[201] 98.84 0.09 1.29 97.80 98.80 99.85 223 1.02
## mu[202] 149.28 0.26 2.98 148.18 149.65 150.26 135 1.04
## mu[203] 94.86 0.09 1.36 93.78 94.79 95.88 231 1.02
## mu[204] 159.45 0.07 1.18 158.72 159.51 160.21 272 1.02
## mu[205] 150.68 0.27 2.85 149.56 151.03 151.62 115 1.04
## mu[206] 161.45 0.05 1.07 160.59 161.39 162.32 386 1.01
## mu[207] 148.79 0.26 3.02 147.67 149.17 149.78 136 1.04
## mu[208] 153.96 0.16 2.03 152.95 154.12 154.94 170 1.03
## mu[209] 154.06 0.29 2.33 153.06 154.34 154.93 65 1.04
## mu[210] 154.27 0.15 2.00 153.24 154.43 155.24 170 1.03
## mu[211] 137.59 0.28 3.35 136.38 137.99 138.66 146 1.04
## mu[212] 155.51 0.20 1.96 154.56 155.75 156.38 95 1.04
## mu[213] 119.29 0.22 2.83 118.08 119.61 120.29 161 1.04
## mu[214] 155.43 0.14 1.87 154.44 155.57 156.37 174 1.03
## mu[215] 161.24 0.04 0.91 160.46 161.22 162.02 449 1.00
## mu[216] 151.54 0.28 2.74 150.43 151.89 152.47 99 1.04
## mu[217] 110.52 0.11 1.47 109.51 110.57 111.54 182 1.02
## mu[218] 161.39 0.11 1.82 160.22 161.22 162.51 289 1.03
## mu[219] 154.50 0.27 2.23 153.50 154.77 155.36 68 1.04
## mu[220] 154.33 0.28 2.27 153.31 154.61 155.20 66 1.04
## mu[221] 100.91 0.09 1.28 99.89 100.90 101.93 215 1.02
## mu[222] 114.70 0.12 1.63 113.74 114.79 115.74 175 1.03
## mu[223] 149.65 0.18 2.31 148.64 149.83 150.75 166 1.04
## mu[224] 159.16 0.08 1.25 158.39 159.23 159.94 245 1.02
## mu[225] 160.62 0.05 0.91 159.88 160.64 161.34 383 1.00
## mu[226] 154.91 0.25 2.13 153.91 155.18 155.78 76 1.04
## mu[227] 119.45 0.14 1.82 118.46 119.58 120.55 170 1.03
## mu[228] 160.73 0.04 0.89 159.99 160.75 161.47 403 1.00
## mu[229] 151.46 0.28 2.75 150.36 151.81 152.39 100 1.04
## mu[230] 148.50 0.26 3.04 147.37 148.87 149.49 136 1.04
## mu[231] 152.32 0.29 2.64 151.30 152.65 153.23 85 1.04
## mu[232] 106.25 0.14 2.17 105.13 106.47 107.19 233 1.03
## mu[233] 148.29 0.26 3.06 147.16 148.67 149.29 137 1.04
## mu[234] 160.70 0.04 0.89 159.96 160.71 161.44 397 1.00
## mu[235] 149.09 0.26 3.00 147.97 149.46 150.07 135 1.04
## mu[236] 154.82 0.15 1.94 153.82 154.97 155.77 172 1.03
## mu[237] 156.73 0.13 1.50 155.64 156.90 157.65 130 1.04
## mu[238] 161.43 0.10 1.73 160.30 161.27 162.53 294 1.03
## mu[239] 154.87 0.25 2.14 153.86 155.14 155.73 75 1.04
## mu[240] 151.05 0.27 2.80 149.92 151.40 151.99 108 1.04
## mu[241] 129.92 0.27 3.22 128.64 130.29 130.99 148 1.04
## mu[242] 149.66 0.26 2.95 148.55 150.02 150.62 134 1.04
## mu[243] 114.11 0.19 2.58 112.87 114.40 115.10 177 1.04
## mu[244] 157.74 0.11 1.54 156.89 157.84 158.59 191 1.03
## mu[245] 138.99 0.28 3.35 137.77 139.39 140.05 145 1.04
## mu[246] 119.67 0.14 1.83 118.68 119.80 120.77 170 1.03
## mu[247] 159.30 0.08 1.22 158.56 159.36 160.07 259 1.02
## mu[248] 148.05 0.26 3.08 146.96 148.43 149.05 137 1.04
## mu[249] 101.30 0.12 1.91 100.20 101.50 102.25 267 1.03
## mu[250] 145.52 0.27 3.22 144.32 145.93 146.55 141 1.04
## mu[251] 151.16 0.27 2.79 150.04 151.51 152.10 106 1.04
## mu[252] 160.54 0.05 0.92 159.81 160.56 161.26 371 1.00
## mu[253] 160.46 0.05 0.94 159.73 160.49 161.18 359 1.01
## mu[254] 155.31 0.22 2.02 154.37 155.56 156.19 88 1.04
## mu[255] 155.96 0.14 1.81 154.99 156.09 156.88 176 1.03
## mu[256] 146.21 0.27 3.18 145.02 146.62 147.23 140 1.04
## mu[257] 157.65 0.11 1.55 156.80 157.75 158.51 190 1.03
## mu[258] 160.63 0.05 0.90 159.89 160.65 161.35 385 1.00
## mu[259] 146.36 0.27 3.18 145.18 146.77 147.37 140 1.04
## mu[260] 160.94 0.16 2.44 159.58 160.71 162.31 240 1.03
## mu[261] 148.83 0.26 3.02 147.70 149.20 149.81 136 1.04
## mu[262] 160.36 0.05 0.96 159.63 160.40 161.06 346 1.01
## mu[263] 145.68 0.27 3.21 144.48 146.09 146.70 141 1.04
## mu[264] 96.04 0.09 1.66 95.06 96.18 97.01 313 1.02
## mu[265] 151.54 0.28 2.74 150.43 151.89 152.47 99 1.04
## mu[266] 98.96 0.11 1.79 97.91 99.14 99.92 285 1.03
## mu[267] 154.35 0.28 2.27 153.34 154.63 155.22 66 1.04
## mu[268] 161.29 0.04 0.93 160.49 161.26 162.07 439 1.00
## mu[269] 161.27 0.12 2.03 160.03 161.08 162.50 279 1.03
## mu[270] 147.92 0.26 3.09 146.83 148.30 148.91 138 1.04
## mu[271] 149.21 0.18 2.33 148.20 149.39 150.33 166 1.04
## mu[272] 160.21 0.21 3.12 158.57 159.90 161.72 217 1.04
## mu[273] 116.70 0.21 2.71 115.49 117.01 117.70 167 1.04
## mu[274] 154.12 0.29 2.32 153.11 154.40 154.99 65 1.04
## mu[275] 158.62 0.09 1.37 157.81 158.70 159.45 213 1.02
## mu[276] 154.22 0.28 2.30 153.20 154.50 155.09 66 1.04
## mu[277] 158.77 0.09 1.33 157.96 158.85 159.59 219 1.02
## mu[278] 150.77 0.27 2.84 149.63 151.12 151.71 113 1.04
## mu[279] 153.88 0.29 2.37 152.86 154.17 154.76 66 1.04
## mu[280] 123.87 0.24 3.02 122.65 124.21 124.91 154 1.04
## mu[281] 143.11 0.28 3.30 141.87 143.52 144.16 143 1.04
## mu[282] 140.51 0.28 3.34 139.28 140.92 141.57 145 1.04
## mu[283] 145.33 0.27 3.22 144.11 145.74 146.36 141 1.04
## mu[284] 161.09 0.04 0.87 160.31 161.08 161.86 455 1.00
## mu[285] 161.51 0.08 1.44 160.50 161.40 162.49 321 1.02
## mu[286] 151.36 0.17 2.22 150.36 151.54 152.43 167 1.03
## mu[287] 150.80 0.27 2.83 149.67 151.15 151.74 113 1.04
## mu[288] 161.32 0.05 0.95 160.51 161.29 162.11 432 1.00
## mu[289] 158.32 0.10 1.43 157.50 158.40 159.15 203 1.03
## mu[290] 142.15 0.28 3.32 140.93 142.56 143.20 144 1.04
## mu[291] 116.41 0.21 2.69 115.20 116.72 117.41 168 1.04
## mu[292] 103.34 0.13 2.01 102.29 103.56 104.29 250 1.03
## mu[293] 161.17 0.04 0.89 160.40 161.16 161.94 455 1.00
## mu[294] 141.84 0.28 3.32 140.61 142.25 142.89 144 1.04
## mu[295] 158.23 0.10 1.44 157.41 158.32 159.07 201 1.03
## mu[296] 156.11 0.17 1.76 155.18 156.32 156.99 107 1.04
## mu[297] 152.85 0.29 2.56 151.81 153.18 153.75 77 1.04
## mu[298] 148.26 0.18 2.36 147.24 148.46 149.39 165 1.04
## mu[299] 123.14 0.24 2.99 121.92 123.48 124.17 155 1.04
## mu[300] 142.33 0.28 3.31 141.11 142.74 143.38 144 1.04
## mu[301] 88.83 0.06 1.43 87.83 88.91 89.92 484 1.01
## mu[302] 151.54 0.28 2.74 150.43 151.89 152.47 99 1.04
## mu[303] 154.49 0.15 1.98 153.47 154.64 155.45 171 1.03
## mu[304] 145.33 0.27 3.22 144.11 145.74 146.36 141 1.04
## mu[305] 161.37 0.05 0.99 160.55 161.33 162.17 417 1.01
## mu[306] 154.96 0.24 2.12 154.00 155.23 155.83 77 1.04
## mu[307] 125.82 0.25 3.09 124.58 126.18 126.87 151 1.04
## mu[308] 99.31 0.11 1.81 98.26 99.49 100.27 282 1.03
## mu[309] 156.47 0.15 1.61 155.49 156.67 157.38 112 1.04
## mu[310] 160.03 0.22 3.26 158.35 159.71 161.57 214 1.04
## mu[311] 153.42 0.29 2.46 152.40 153.73 154.31 70 1.04
## mu[312] 111.26 0.11 1.50 110.25 111.31 112.28 180 1.02
## mu[313] 161.49 0.06 1.16 160.61 161.42 162.39 362 1.02
## mu[314] 161.34 0.05 0.96 160.53 161.30 162.13 428 1.01
## mu[315] 142.85 0.28 3.30 141.62 143.26 143.90 144 1.04
## mu[316] 149.60 0.26 2.95 148.49 149.96 150.56 134 1.04
## mu[317] 159.64 0.24 3.54 157.85 159.29 161.26 209 1.04
## mu[318] 155.11 0.15 1.91 154.11 155.26 156.06 173 1.03
## mu[319] 161.50 0.06 1.20 160.60 161.43 162.42 353 1.02
## mu[320] 150.24 0.26 2.89 149.08 150.60 151.19 124 1.04
## mu[321] 158.30 0.10 1.43 157.48 158.38 159.13 203 1.03
## mu[322] 156.87 0.13 1.61 155.40 156.91 158.17 156 1.04
## mu[323] 154.55 0.27 2.22 153.58 154.83 155.42 69 1.04
## mu[324] 91.25 0.09 1.49 90.13 91.15 92.30 249 1.02
## mu[325] 92.30 0.09 1.45 91.21 92.21 93.34 242 1.02
## mu[326] 66.57 0.10 2.19 65.06 66.45 68.29 449 1.02
## mu[327] 93.45 0.09 1.41 92.37 93.36 94.48 235 1.02
## mu[328] 108.12 0.15 2.27 106.97 108.36 109.06 217 1.03
## mu[329] 86.44 0.06 1.40 85.41 86.51 87.58 594 1.01
## mu[330] 151.60 0.28 2.74 150.48 151.94 152.52 98 1.04
## mu[331] 102.07 0.12 1.95 100.98 102.27 103.02 260 1.03
## mu[332] 70.04 0.08 1.95 68.64 69.95 71.65 541 1.02
## mu[333] 92.69 0.09 1.43 91.60 92.60 93.73 240 1.02
## mu[334] 98.17 0.10 1.75 97.13 98.34 99.14 292 1.02
## mu[335] 85.75 0.12 1.78 84.52 85.58 86.99 216 1.02
## mu[336] 152.12 0.28 2.67 151.06 152.45 153.03 89 1.04
## mu[337] 157.18 0.12 1.63 156.29 157.30 158.06 184 1.03
## mu[338] 137.96 0.19 2.42 136.99 138.14 139.11 164 1.04
## mu[339] 91.19 0.07 1.48 90.20 91.28 92.23 405 1.01
## mu[340] 157.69 0.11 1.55 156.84 157.80 158.55 190 1.03
## mu[341] 151.11 0.27 2.80 149.98 151.46 152.04 107 1.04
## mu[342] 157.46 0.12 1.58 156.60 157.57 158.34 187 1.03
## mu[343] 155.22 0.22 2.05 154.25 155.47 156.10 84 1.04
## mu[344] 114.33 0.20 2.59 113.11 114.63 115.32 176 1.04
## mu[345] 100.96 0.12 1.89 99.86 101.15 101.90 270 1.03
## mu[346] 160.52 0.05 0.93 159.78 160.54 161.25 367 1.00
## mu[347] 152.66 0.29 2.59 151.64 152.99 153.56 80 1.04
## mu[348] 104.51 0.13 2.07 103.46 104.73 105.47 243 1.03
## mu[349] 97.00 0.09 1.31 95.93 96.95 98.02 226 1.02
## mu[350] 160.76 0.04 0.88 160.02 160.78 161.50 408 1.00
## mu[351] 159.55 0.07 1.16 158.82 159.60 160.31 277 1.02
## mu[352] 161.52 0.08 1.37 160.53 161.42 162.47 329 1.02
## mu[353] 160.40 0.05 0.95 159.67 160.43 161.09 351 1.01
## mu[354] 91.65 0.08 1.50 90.67 91.75 92.70 393 1.01
## mu[355] 70.21 0.20 2.96 68.53 69.89 71.93 209 1.03
## mu[356] 138.07 0.19 2.42 137.10 138.26 139.22 164 1.04
## mu[357] 153.48 0.29 2.45 152.47 153.79 154.37 69 1.04
## mu[358] 153.39 0.29 2.46 152.38 153.70 154.29 70 1.04
## mu[359] 91.44 0.09 1.48 90.33 91.34 92.50 248 1.02
## mu[360] 87.59 0.06 1.41 86.58 87.67 88.70 538 1.01
## mu[361] 77.32 0.19 2.38 75.85 77.08 78.91 155 1.03
## mu[362] 90.91 0.07 1.48 89.91 91.00 91.96 413 1.01
## mu[363] 71.53 0.07 1.85 70.17 71.47 73.11 611 1.01
## mu[364] 94.30 0.09 1.38 93.23 94.22 95.32 233 1.02
## mu[365] 73.75 0.07 1.72 72.42 73.71 75.24 700 1.01
## mu[366] 157.00 0.11 1.44 155.67 157.08 158.15 159 1.04
## mu[367] 154.29 0.28 2.28 153.27 154.57 155.16 66 1.04
## mu[368] 94.96 0.09 1.36 93.87 94.89 95.97 231 1.02
## mu[369] 95.42 0.09 1.35 94.33 95.36 96.44 229 1.02
## mu[370] 72.17 0.07 1.81 70.82 72.12 73.73 638 1.01
## mu[371] 78.91 0.05 1.49 77.70 78.92 80.25 872 1.01
## mu[372] 160.23 0.05 0.99 159.51 160.27 160.94 332 1.01
## mu[373] 133.00 0.27 3.30 131.75 133.38 134.07 147 1.04
## mu[374] 138.64 0.28 3.35 137.43 139.05 139.71 146 1.04
## mu[375] 160.97 0.15 2.40 159.62 160.75 162.33 242 1.03
## mu[376] 117.34 0.13 1.74 116.35 117.45 118.42 172 1.03
## mu[377] 80.87 0.17 2.11 79.50 80.67 82.28 163 1.03
## mu[378] 75.63 0.06 1.63 74.35 75.60 77.07 768 1.01
## mu[379] 87.31 0.06 1.40 86.28 87.38 88.41 551 1.01
## mu[380] 161.15 0.14 2.19 159.87 160.94 162.46 261 1.03
## mu[381] 74.38 0.06 1.69 73.07 74.35 75.85 724 1.01
## mu[382] 130.10 0.27 3.23 128.81 130.47 131.17 148 1.04
## mu[383] 160.36 0.05 0.96 159.63 160.40 161.06 346 1.01
## mu[384] 89.01 0.10 1.60 87.83 88.89 90.15 261 1.02
## mu[385] 95.13 0.09 1.62 94.17 95.27 96.13 325 1.02
## mu[386] 154.60 0.26 2.21 153.66 154.88 155.47 70 1.04
## mu[387] 154.83 0.25 2.15 153.83 155.10 155.69 74 1.04
## mu[388] 156.37 0.16 1.65 155.37 156.57 157.26 110 1.04
## mu[389] 159.93 0.06 1.06 159.20 159.98 160.66 307 1.01
## mu[390] 113.12 0.19 2.53 111.89 113.40 114.11 182 1.04
## mu[391] 112.15 0.11 1.53 111.15 112.21 113.19 179 1.03
## mu[392] 89.40 0.10 1.58 88.25 89.29 90.54 259 1.02
## mu[393] 153.74 0.29 2.40 152.72 154.03 154.62 67 1.04
## mu[394] 134.50 0.18 2.35 133.53 134.66 135.64 165 1.04
## mu[395] 114.70 0.12 1.63 113.74 114.79 115.74 175 1.03
## mu[396] 94.59 0.09 1.60 93.63 94.72 95.58 333 1.02
## mu[397] 161.09 0.14 2.27 159.78 160.87 162.41 253 1.03
## mu[398] 150.36 0.26 2.88 149.20 150.72 151.31 121 1.04
## mu[399] 133.23 0.18 2.32 132.26 133.41 134.37 165 1.04
## mu[400] 156.84 0.12 1.45 155.69 157.00 157.80 152 1.04
## mu[401] 83.43 0.14 1.93 82.14 83.24 84.73 185 1.03
## mu[402] 77.43 0.19 2.37 75.96 77.18 79.02 155 1.03
## mu[403] 159.78 0.06 1.10 159.05 159.84 160.52 294 1.01
## mu[404] 111.20 0.17 2.43 110.01 111.47 112.15 192 1.03
## mu[405] 152.40 0.17 2.15 151.39 152.57 153.44 168 1.03
## mu[406] 161.03 0.04 0.87 160.25 161.03 161.79 451 1.00
## mu[407] 71.96 0.07 1.83 70.60 71.90 73.52 630 1.01
## mu[408] 136.55 0.19 2.39 135.58 136.72 137.68 164 1.04
## mu[409] 78.37 0.18 2.30 76.93 78.14 79.90 157 1.03
## mu[410] 153.90 0.29 2.36 152.88 154.19 154.78 65 1.04
## mu[411] 161.39 0.05 1.00 160.56 161.35 162.19 411 1.01
## mu[412] 155.57 0.14 1.86 154.58 155.71 156.50 174 1.03
## mu[413] 147.46 0.26 3.12 146.36 147.85 148.46 138 1.04
## mu[414] 161.46 0.06 1.09 160.60 161.40 162.34 380 1.01
## mu[415] 154.39 0.28 2.26 153.37 154.66 155.26 67 1.04
## mu[416] 74.58 0.06 1.68 73.28 74.55 76.05 731 1.01
## mu[417] 153.37 0.29 2.47 152.36 153.68 154.27 70 1.04
## mu[418] 160.00 0.06 1.05 159.28 160.05 160.73 312 1.01
## mu[419] 149.78 0.25 2.94 148.67 150.15 150.74 133 1.04
## mu[420] 143.66 0.27 3.28 142.42 144.07 144.70 143 1.04
## mu[421] 124.51 0.16 2.03 123.53 124.66 125.60 167 1.03
## mu[422] 154.02 0.16 2.02 153.01 154.18 155.00 170 1.03
## mu[423] 147.32 0.27 3.12 146.21 147.70 148.31 139 1.04
## mu[424] 160.82 0.04 0.88 160.07 160.83 161.56 419 1.00
## mu[425] 144.85 0.27 3.24 143.61 145.26 145.89 142 1.04
## mu[426] 84.04 0.14 1.89 82.75 83.86 85.33 192 1.03
## mu[427] 158.13 0.10 1.47 157.31 158.22 158.96 199 1.03
## mu[428] 160.65 0.05 0.90 159.91 160.67 161.38 388 1.00
## mu[429] 139.87 0.19 2.44 138.89 140.07 141.04 164 1.04
## mu[430] 161.38 0.05 0.99 160.56 161.34 162.18 414 1.01
## mu[431] 153.82 0.29 2.38 152.79 154.11 154.70 66 1.04
## mu[432] 136.83 0.19 2.40 135.87 137.01 137.97 164 1.04
## mu[433] 118.25 0.14 1.77 117.26 118.37 119.34 171 1.03
## mu[434] 160.65 0.05 0.90 159.91 160.67 161.38 388 1.00
## mu[435] 153.89 0.16 2.03 152.89 154.06 154.88 170 1.03
## mu[436] 159.99 0.06 1.05 159.27 160.04 160.72 311 1.01
## mu[437] 160.98 0.15 2.39 159.63 160.76 162.34 243 1.03
## mu[438] 160.79 0.04 0.88 160.04 160.80 161.53 413 1.00
## mu[439] 156.14 0.17 1.74 155.21 156.35 157.03 107 1.04
## mu[440] 97.37 0.09 1.31 96.31 97.33 98.39 225 1.02
## mu[441] 69.50 0.09 1.98 68.10 69.41 71.13 522 1.02
## mu[442] 160.36 0.05 0.96 159.63 160.40 161.06 346 1.01
## mu[443] 144.77 0.27 3.25 143.53 145.18 145.81 142 1.04
## mu[444] 146.21 0.27 3.18 145.02 146.62 147.23 140 1.04
## mu[445] 156.40 0.16 1.64 155.39 156.59 157.29 111 1.04
## mu[446] 161.42 0.05 1.04 160.57 161.37 162.25 399 1.01
## mu[447] 159.90 0.06 1.07 159.17 159.95 160.63 304 1.01
## mu[448] 146.70 0.27 3.16 145.54 147.10 147.70 140 1.04
## mu[449] 130.02 0.17 2.22 129.05 130.19 131.14 165 1.03
## mu[450] 106.68 0.10 1.36 105.62 106.71 107.70 192 1.02
## mu[451] 160.81 0.04 0.88 160.06 160.82 161.55 417 1.00
## mu[452] 148.05 0.26 3.08 146.96 148.43 149.05 137 1.04
## mu[453] 76.35 0.06 1.59 75.08 76.33 77.78 801 1.01
## mu[454] 160.27 0.05 0.98 159.55 160.31 160.98 336 1.01
## mu[455] 132.43 0.27 3.28 131.16 132.81 133.50 147 1.04
## mu[456] 75.00 0.06 1.66 73.71 74.97 76.46 746 1.01
## mu[457] 76.05 0.19 2.48 74.55 75.79 77.65 163 1.03
## mu[458] 161.24 0.12 2.06 160.00 161.05 162.49 276 1.03
## mu[459] 144.56 0.19 2.44 143.60 144.76 145.71 165 1.04
## mu[460] 161.44 0.10 1.71 160.33 161.29 162.53 296 1.03
## mu[461] 160.75 0.04 0.88 160.01 160.77 161.49 406 1.00
## mu[462] 153.11 0.29 2.51 152.06 153.42 154.00 74 1.04
## mu[463] 71.74 0.07 1.84 70.38 71.69 73.32 621 1.01
## mu[464] 142.77 0.28 3.30 141.53 143.18 143.81 144 1.04
## mu[465] 156.67 0.14 1.52 155.60 156.85 157.59 123 1.04
## mu[466] 156.09 0.13 1.79 155.14 156.23 157.01 177 1.03
## mu[467] 62.48 0.13 2.51 60.81 62.31 64.38 378 1.02
## mu[468] 72.40 0.19 2.78 70.78 72.09 74.06 209 1.03
## mu[469] 154.06 0.29 2.33 153.06 154.34 154.93 65 1.04
## mu[470] 141.62 0.28 3.33 140.38 142.02 142.67 144 1.04
## mu[471] 104.01 0.13 2.05 102.96 104.23 104.96 246 1.03
## mu[472] 75.31 0.06 1.64 74.04 75.28 76.76 758 1.01
## mu[473] 160.27 0.05 0.98 159.55 160.31 160.98 336 1.01
## mu[474] 155.17 0.23 2.06 154.20 155.43 156.05 83 1.04
## mu[475] 113.28 0.19 2.53 112.05 113.56 114.26 181 1.04
## mu[476] 95.61 0.09 1.34 94.52 95.55 96.62 228 1.02
## mu[477] 160.32 0.05 0.97 159.60 160.36 161.02 342 1.01
## mu[478] 151.13 0.27 2.80 150.01 151.48 152.07 106 1.04
## mu[479] 161.27 0.04 0.92 160.48 161.25 162.05 442 1.00
## mu[480] 154.59 0.27 2.21 153.63 154.86 155.46 69 1.04
## mu[481] 87.31 0.06 1.40 86.28 87.38 88.41 551 1.01
## mu[482] 156.94 0.12 1.67 156.03 157.06 157.83 182 1.03
## mu[483] 151.89 0.28 2.70 150.78 152.23 152.81 93 1.04
## mu[484] 83.23 0.14 1.94 81.93 83.03 84.53 183 1.03
## mu[485] 159.16 0.08 1.25 158.39 159.23 159.94 245 1.02
## mu[486] 156.90 0.11 1.42 155.76 157.05 157.89 157 1.04
## mu[487] 120.49 0.14 1.87 119.47 120.62 121.58 170 1.03
## mu[488] 91.73 0.09 1.47 90.62 91.63 92.78 246 1.02
## mu[489] 161.01 0.04 0.87 160.23 161.01 161.76 448 1.00
## mu[490] 131.62 0.27 3.27 130.34 132.00 132.70 148 1.04
## mu[491] 161.42 0.05 1.03 160.57 161.36 162.24 402 1.01
## mu[492] 153.71 0.29 2.40 152.71 154.01 154.60 67 1.04
## mu[493] 110.85 0.11 1.48 109.83 110.90 111.87 181 1.02
## mu[494] 144.73 0.27 3.25 143.49 145.14 145.77 142 1.04
## mu[495] 142.85 0.28 3.30 141.62 143.26 143.90 144 1.04
## mu[496] 152.96 0.29 2.54 151.90 153.28 153.86 76 1.04
## mu[497] 140.14 0.28 3.34 138.90 140.54 141.20 145 1.04
## mu[498] 160.03 0.06 1.04 159.31 160.08 160.76 314 1.01
## mu[499] 156.61 0.13 1.72 155.69 156.74 157.51 180 1.03
## mu[500] 152.76 0.16 2.13 151.76 152.93 153.79 168 1.03
## mu[501] 157.81 0.11 1.52 156.97 157.92 158.66 192 1.03
## mu[502] 161.49 0.09 1.55 160.43 161.36 162.51 308 1.03
## mu[503] 153.04 0.29 2.52 151.97 153.36 153.93 75 1.04
## mu[504] 121.37 0.15 1.90 120.35 121.51 122.45 169 1.03
## mu[505] 86.63 0.06 1.40 85.61 86.70 87.76 584 1.01
## mu[506] 154.55 0.15 1.97 153.54 154.70 155.51 171 1.03
## mu[507] 153.33 0.29 2.47 152.31 153.64 154.22 71 1.04
## mu[508] 99.75 0.11 1.83 98.69 99.93 100.70 279 1.03
## mu[509] 122.31 0.15 1.94 121.30 122.46 123.39 168 1.03
## mu[510] 161.52 0.07 1.31 160.56 161.43 162.45 336 1.02
## mu[511] 148.53 0.26 3.04 147.40 148.90 149.52 136 1.04
## mu[512] 110.81 0.17 2.41 109.61 111.08 111.76 195 1.03
## mu[513] 90.09 0.10 1.54 88.95 89.98 91.18 256 1.02
## mu[514] 93.50 0.08 1.56 92.54 93.61 94.50 352 1.02
## mu[515] 152.90 0.29 2.55 151.85 153.22 153.80 77 1.04
## mu[516] 145.25 0.27 3.23 144.03 145.66 146.28 141 1.04
## mu[517] 153.98 0.29 2.35 152.97 154.27 154.86 65 1.04
## mu[518] 60.12 0.14 2.70 58.35 59.93 62.10 348 1.02
## mu[519] 148.09 0.26 3.07 146.99 148.47 149.08 137 1.04
## mu[520] 63.71 0.12 2.41 62.08 63.55 65.52 397 1.02
## mu[521] 100.64 0.09 1.28 99.63 100.63 101.67 216 1.02
## mu[522] 125.55 0.16 2.07 124.57 125.71 126.64 166 1.03
## mu[523] 84.00 0.05 1.40 82.95 84.05 85.19 724 1.00
## mu[524] 158.42 0.10 1.41 157.61 158.50 159.25 206 1.02
## mu[525] 153.84 0.29 2.38 152.81 154.13 154.72 66 1.04
## mu[526] 129.70 0.17 2.21 128.72 129.87 130.81 165 1.03
## mu[527] 119.50 0.22 2.84 118.29 119.82 120.51 161 1.04
## mu[528] 96.21 0.09 1.67 95.23 96.36 97.19 311 1.02
## mu[529] 160.53 0.05 0.92 159.79 160.55 161.25 369 1.00
## mu[530] 149.63 0.26 2.95 148.52 149.99 150.59 134 1.04
## mu[531] 124.72 0.25 3.05 123.50 125.07 125.77 153 1.04
## mu[532] 84.00 0.05 1.40 82.95 84.05 85.19 724 1.00
## mu[533] 155.85 0.14 1.82 154.88 155.99 156.77 175 1.03
## mu[534] 151.11 0.27 2.80 149.98 151.46 152.04 107 1.04
## mu[535] 155.63 0.19 1.92 154.70 155.87 156.50 99 1.04
## mu[536] 110.36 0.11 1.47 109.34 110.41 111.37 182 1.02
## mu[537] 71.64 0.07 1.85 70.28 71.58 73.22 616 1.01
## mu[538] 142.85 0.28 3.30 141.62 143.26 143.90 144 1.04
## mu[539] 79.10 0.18 2.24 77.68 78.88 80.63 158 1.03
## mu[540] 142.21 0.19 2.45 141.25 142.42 143.37 164 1.04
## mu[541] 161.50 0.06 1.20 160.60 161.43 162.42 353 1.02
## mu[542] 156.94 0.12 1.53 155.51 157.00 158.17 157 1.04
## mu[543] 79.21 0.18 2.23 77.78 78.99 80.74 158 1.03
## mu[544] 161.52 0.07 1.28 160.58 161.43 162.44 340 1.02
## lp__ -1323.61 164.14 238.94 -1685.71 -1190.02 -1188.07 2 4.82
##
## Samples were drawn using NUTS(diag_e) at Fri Nov 22 17:59:50 2019.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).